THE WIENER CRITERION FOR FULLY NONLINEAR ELLIPTIC EQUATIONS

Abstract

We study the boundary continuity of solutions to fully nonlinear elliptic equations. We first define a capacity for operators in non-divergence form and derive several capacitary estimates. Secondly, we formulate the Wiener criterion, which characterizes a regular boundary point via potential theory. Our approach utilizes the asymptotic behavior of homogeneous solutions, together with Harnack inequality and the comparison principle.

1. Introduction

Let Ω be an open and bounded subset in ${ℝ}^n$, f be a boundary data on ∂Ω, and ℳ be an elliptic operator. For the existence of a solution u (in a suitable sense) to the Dirichlet problem

$$\{\begin{array}{ll}ℳ[u]=0\hfill & \text{in }\Omega ,\hfill \\ u=f\hfill & \text{on }\partial \Omega ,\hfill \end{array}$$

one may apply Perron’s method. If the solvability of the Dirichlet problem on any balls is known and ℳ allows a comparison principle, it is rather straightforward to prove that the upper Perron solution ${\overline{H}}_f$ satisfies $ℳ\left[{\overline{H}}_f\right]=0$ in Ω. (See Section 2 for details.) Nevertheless, we cannot ensure that the boundary condition u = f on ∂Ω is satisfied by the upper Perron solution, in general. Instead, we are forced to discover an additional condition for the boundary ∂Ω, which enables us to capture the boundary behavior of ${\overline{H}}_f$.

To be precise, we say a boundary point x₀ ∈ ∂Ω is regular with respect to Ω, if

$$\underset{\Omega \ni y\to x_0}{\text{lim}}{\overline{H}}_f(y)=f\left(x_0\right).$$

whenever f ∈ C(∂Ω). One simple characterization of a regular boundary point is to find a barrier function; see Section 2 for the precise definition. As a consequence, by constructing proper barrier functions, geometric criteria on ∂Ω such as an exterior sphere condition or an exterior cone condition have been invoked to guarantee the boundary continuity at x₀ ∈ ∂Ω for a variety of elliptic operators.

On the other hand, Wiener [40] developed an alternative criterion for a regular boundary point, based on potential theory. Namely, for the Laplacian operator (ℳ = Δ), x₀ ∈ ∂Ω is regular if and only if the Wiener integral diverges, i.e.

$${\int }_0^1\frac{{\text{cap}}_2\left(\overline{B_t\left(x_0\right)}\backslash \Omega ,B_{2t}\left(x_0\right)\right)}{{\text{cap}}_2\left(\overline{B_t\left(x_0\right)},B_{2t}\left(x_0\right)\right)}\frac{\text{d}t}{t}=\infty ,$$

where cap₂(K, Ω) is defined by the variational capacity of the Laplacian operator. Surprisingly, the Wiener criterion becomes both a sufficient and necessary condition for the regularity of a boundary point. Here the notion of capacity is used to measure the ‘size’ of sets in view of given differential equations. Roughly speaking, x₀ ∈ ∂Ω is regular if and only if Ω^c is ‘thick’ enough at x₀ in the potential theoretic sense.

Both linear and nonlinear potential theory have been extensively studied in literature; see [5, 12, 13, 25, 31, 39] and references therein. Since the main ingredient of potential theory comes from the integration by parts, the theory and corresponding Wiener criterion have been developed mostly for operators in divergence form. Littman, Stampacchia and Weinburger [30] demonstrated the coincidence between the regular points for uniformly elliptic operators ℳ = −D_j(a_ijD_i), where a_ij is bounded and measurable, and for the Laplacian operator. For the p-Laplacian operator (ℳ = Δ_p, p > 1), Maz’ya [32] verified the sufficiency of the p-Wiener criterion, i.e. x₀ ∈ ∂Ω is regular for Δ_p if

$${\int }_0^1{\left(\frac{{\text{cap}}_p\left(\overline{B_t\left(x_0\right)}\backslash \Omega ,B_{2t}\left(x_0\right)\right)}{{\text{cap}}_p\left(\overline{B_t\left(x_0\right)},B_{2t}\left(x_0\right)\right)}\right)}^{1/(p-1)}\frac{\text{d}t}{t}=\infty .$$

For the converse direction, Lindqvist and Martio [29] proved the necessity of the Wiener criterion under the assumption p > n − 1. Later, Kilpeläinen and Malý [20] extended this result to any p > 1, via the Wolff potential estimate. For the other available results on the Wiener criterion, we refer to [1] for p(x)-Laplacian operators and [27] for operators with Orlicz growth. Note that all of these results consider elliptic operators in divergence form.

For elliptic operators in non-divergence form, relatively small amounts of results for the Wiener criterion are known. While the equivalence was obtained for ℳ = D_j(a_ijD_i) with merely measurable coefficients in [30], Miller [35, 36] discovered the non-equivalence with respect to ℳ = a_ijD_iju, even if the coefficients a_ij are continuous. More precisely, he presented examples of linear operators ℳ in non-divergence form and domains Ω such that x₀ ∈ ∂Ω is regular for ℳ, but x₀ is irregular for Δ, and vice-versa. We also refer [22, 26]. On the other hand, Bauman [4] developed the Wiener test for ℳ = a_ijD_iju with continuous coefficients a_ij. He proved that x₀ ∈ ∂Ω is regular if and only if

1. cap_ℳ({x₀}) > 0,

2. ${\sum }_{j=1}^{\infty }\tilde{g}\left(x_0,x_0+2^{-j}e\right)\cdot {\text{cap}}_{ℳ}\left({\Omega }^c\cap \left(\overline{B_{2^{-j}}\left(x_0\right)}\backslash B_{2^{-j-1}}\left(x_0\right)\right)\right)=\infty$.

Here $\tilde{g}$ is the normalized Green function and e is a unit vector in ${ℝ}^n$.

The goal of this paper is to establish the Wiener criterion for fully nonlinear elliptic operators, by implementing potential theoretic tools. To illustrate the issues, we consider an Issacs operator, i.e. an operator F with the following two properties:

• (F1) F is uniformly elliptic: there exist positive constants 0 < λ ≤ Λ such that for any M ∈ 𝒮ⁿ,

  $$\lambda \Vert N\Vert \le F(M+N)-F(M)\le \Lambda \Vert N\Vert ,\forall N\ge 0.$$

  Here we write N ≥ 0 whenever N is a non-negative definite symmetric matrix.

• (F2) F is positively homogeneous of degree one: F(tM) = tF(M) for any t > 0 and M ∈ 𝒮ⁿ.

Throughout the present paper, we suppose that F satisfies (F1) and (F2), unless otherwise stated. Typical examples of operators satisfying (F1) and (F2) are the Pucci extremal operators ${𝒫}_{\lambda ,\Lambda }^{+}$ and ${𝒫}_{\lambda ,\Lambda }^{-}$, defined by

$${𝒫}_{\lambda ,\Lambda }^{+}(M)=\Lambda \sum _{e_i>0}{e}_i+\lambda \sum _{e_i<0}{e}_i,{𝒫}_{\lambda ,\Lambda }^{-}(M)=\Lambda \sum _{e_i<0}{e}_i+\lambda \sum _{e_i>0}{e}_i,$$

where e_i = e_i(M) are the eigenvalues of M. For a fully nonlinear operator F satisfying (F1) and (F2), we define a dual operator

$$\tilde{F}(M)≔-F(-M),\text{ for }M\in {𝒮}^n.$$

Then it is obvious that $\tilde{F}$ also satisfies (F1) and (F2). One important property that F satisfying (F1) and (F2) possesses is the existence of a homogeneous solution V :

Lemma 1.1 (A homogeneous solution; [3, 7]). There exists a non-constant solution of F(D²u) = 0 in ${ℝ}^n\backslash \left\{0\right\}$ that is bounded below in B₁ and bounded above in ${ℝ}^n\backslash B_1$. Moreover, the set of all such solutions is of the form $\{aV+b\mid a>0,b\in ℝ\}$, where $V\in C_{\text{loc}}^{1,\gamma }\left({ℝ}^n\backslash \left\{0\right\}\right)$ can be chosen to satisfy one of the following homogeneity relations: for all t > 0

$$V(x)=V(tx)+\text{log }t\text{ in }{ℝ}^n\backslash \left\{0\right\}\mathit{\text{ where }}{\alpha }^{*}=0,$$

or

$$V(x)=t^{{\alpha }^{*}}V(tx),{\alpha }^{*}V>0\mathit{\text{ in }}{ℝ}^n\backslash \left\{0\right\},$$

for some number α* ∈ (−1, ∞)\{0} that depends only on F and n. We call the number α* = α*(F) the scaling exponent of F.

Now we are ready to state our first main theorem, namely, the sufficiency of the Wiener criterion:

Theorem 1.2 (The sufficiency of the Wiener crietrion). If

$${\int }_0^1{\text{cap}}_F\left(\overline{B_t\left(x_0\right)}\backslash \Omega ,B_{2t}\left(x_0\right)\right)\frac{\text{d}t}{t}=\infty$$

and

$${\int }_0^1{\text{cap}}_{\tilde{F}}\left(\overline{B_t\left(x_0\right)}\backslash \Omega ,B_{2t}\left(x_0\right)\right)\frac{\text{d}t}{t}=\infty ,$$

then the boundary point x₀ ∈ ∂Ω is (F-)regular.

We remark that the Wiener integral is again defined in terms of a capacity, but the definition of a F-capacity is quite different from the variational capacity for the Laplacian case; see Section 3 for details. Furthermore, as a corollary of Theorem 1.2, we will derive the quantitative estimate for a modulus of continuity at a regular boundary point (Lemma 4.7), and suggest another geometric condition, called an exterior corkscrew condition (Corollary 4.9).

Our second main theorem is concerned with the necessity of the Wiener criterion. We propose a partial result on the necessary condition, i.e. exploiting the additional structure of F, we show that the Wiener integral at x₀ ∈ ∂Ω must diverge whenever x₀ is a regular boundary point.

Theorem 1.3 (The necessity of the Wiener criterion). Suppose that F is concave and α*(F) < 1. If a boundary point x₀ ∈ ∂Ω is regular, then

$${\int }_0^1{\text{cap}}_F\left(\overline{B_t\left(x_0\right)}\backslash \Omega ,B_{2t}\left(x_0\right)\right)\frac{\text{d}t}{t}=\infty .$$

Note that the assumption α*(F) < 1 in the fully nonlinear case corresponds to the assumption p > n − 1 in the p-Laplacian case, [29]. The underlying idea for both cases is to utilize the non-zero capacity of a line segment (or a set of Hausdorff dimension 1). Further comments on this assumption can be found in Section 5.

In this paper, the main difficulty arises from the inherent lack of divergence structure; we cannot define a variational capacity by means of an energy minimizer, and moreover, we cannot employ integral estimates involving Sobolev inequality and Poincaré inequality. Instead, we will develop potential theory with non-divergence structure by the construction of appropriate barrier functions using the homogeneous solution, and by the application of the comparison principle and Harnack inequality. In short, our strategy is to capture the local boundary behavior of the upper Perron solution ${\overline{H}}_f$ in terms of newly defined capacity cap_F (K, B) and the capacity potential (or the balayage) ${\widehat{R}}_K^1(B)$, using prescribed tools. Heuristically, the non-variational capacity measures the ‘height’ of the F-solution with the boundary value 0 on ∂B and 1 on ∂K, while the variational capacity measures the ‘energy’ of such function. We emphasize that although our notion of capacity does not satisfy the subadditive property in general, it was still able to recover certain properties of the variational capacity.

Finally, we would like to point out that the dual operator $\tilde{F}$ is different from F, for general F. Thus, even though u is an F-supersolution, we cannot guarantee −u is an F-subsolution. Moreover, a similar feature is found in the growth rate of the homogeneous solution for F; two growth rates of an upward-pointing homogeneous solution and a downward-pointing one can be different. This phenomenon naturally leads us

1. to describe the local behavior of both the upper Perron solution ${\overline{H}}_f$ and the lower Perron solution ${\underset{\_}{H}}_f$ for regularity at x₀ ∈ ∂Ω;

2. to construct two (upper/lower) barrier functions when characterizing a regular boundary point;

3. to display two different Wiener integrals in our main theorem,

which differ from the previous results that appeared in [4, 20, 40].

Outline.

This paper is organized as follows. In Section 2, we summarize the terminology and preliminary results for our main theorems. In short, we introduce F-superharmonic functions and Poisson modification and then perform Perron’s method. In Section 3, we first define a balayage and a capacity for uniformly elliptic operators in non-divergence form. Then we prove several capacitary estimates by constructing auxiliary functions and provide the characterization of a regular boundary point via balayage. Section 4 consists of potential theoretic estimates for the capacity potential. Then we prove the sufficiency of the Wiener criterion and several corollaries. Finally, Section 5 is devoted to the proof of the (partial) necessity of the Wiener criterion.

2. Perron’s method

2.1. F-supersolutions and F-superharmonic functions.

In this subsection, we only require the condition (F1) for an operator F. To illustrate Perron’s method precisely, we start with two different notions of solutions for a uniformly elliptic operator F: F-solutions and F-harmonic functions. Indeed, we will prove that these two notions coincide.

Definition 2.1 (F-supersolution). A lower semi-continuous [resp. upper semi-continuous] function u in Ω is a (viscosity) F-supersolution [resp. (viscosity) F-subsolution] in Ω, when the following condition holds:

if x₀ ∈ Ω, φ ∈ C²(Ω) and u − φ has a local minimum at x₀, then

$$F\left(D^2\phi \left(x_0\right)\right)\le 0.$$

[resp. if u − φ has a local maximum at x₀, then F(D²φ(x₀)) ≥ 0.]

We say that u ∈ C(Ω) a (viscosity) F-solution if u is both an F-subsolution and an F-supersolution.

Lemma 2.2. Suppose that a lower semi-continuous function u is an F-supersolution in Ω. Then

$$u(x)=\underset{\Omega \ni y\to x}{\text{lim inf }}u(y)\mathit{\text{ for any }}x\in \Omega .$$

Proof. We argue by contradiction: suppose that

$$u\left(x_0\right)<\underset{\Omega \ni y\to x}{\text{lim inf }}u(y)\text{ for some }{x}_0\in \Omega .$$

Then for any φ ∈ C²(Ω), it follows that u − φ has a local minimum at x₀ and so we can test this function. Therefore,

$$F\left(D^2\phi \left(x_0\right)\right)\le 0\text{ for any }\phi \in C^2(\Omega ),$$

which is impossible.

Theorem 2.3.

1. (Stability) Let {u_k}_k≥1 ⊂ C(Ω) be a sequence of F-solutions in Ω. Assume that u_k converges uniformly in every compact set of Ω to u. Then u is an F-solution in Ω.

2. (Compactness) Suppose that {u_k}_k≥1 ⊂ C(Ω) is a locally uniformly bounded sequence of F-solutions in Ω. Then it has a subsequence that converges locally uniformly in Ω to an F-solution.

Theorem 2.4 (Harnack convergence theorem). Let {u_k}_k≥1 ⊂ C(Ω) be an increasing sequence of F-solutions in Ω. Then the function u = lim_k→∞ u_k is either an F-solution or identically +∞ in Ω.

Proof. If u(x) < ∞ for some x ∈ Ω, it follows from Harnack inequality that u is locally bounded in Ω. The interior C^α-estimate yields that the sequence u_k is equicontinuous in every compact subset of Ω. Thus, applying Arzela-Ascoli theorem and Theorem 2.3 (i), we finish the proof. □

We demonstrate two essential tools for Perron’s method, namely, the comparison principle and the solvability of the Dirichlet problem in a ball.

Theorem 2.5 (Comparison principle for F-super/subsolutions, [17, 18]). Let Ω be a bounded open subset of ${ℝ}^n$. Let $v\in \text{USC}(\overline{\Omega })[\mathit{\text{resp}}.u\in \text{LSC}(\overline{\Omega })]$ be an F-subsolution [resp. F-supersolution] in Ω and v ≤ u on ∂Ω. Then v ≤ u in $\overline{\Omega }$.

In the previous theorem, $\text{USC}(\overline{\Omega })$ denotes the set of all upper semi-continuous functions from $\overline{\Omega }$ to $ℝ$. Moreover, note that for a lower semi-continuous function f, there exists an increasing sequence of continuous functions {f_n} such that f_n → f pointwise as n → ∞.

Theorem 2.6 (The solvability of the Dirichlet problem). Let Ω satisfy a uniform exterior cone condition and f ∈ C(∂Ω). Then there exists a unique F-solution $u\in C(\overline{\Omega })$ of the Dirichlet problem

$$\{\begin{array}{ll}F\left(D^{2}u\right)=0\hfill & \text{in }\Omega ,\hfill \\ u=f\hfill & \text{on }\partial \Omega .\hfill \end{array}$$

Proof. The existence depends on the construction of global barriers achieving given boundary data and the standard Perron’s method; see [9, 33] and [8, 15]. Then the uniqueness comes from the comparison principle, Theorem 2.5.

Definition 2.7 (F-superharmonic function). A function u : Ω → (−∞, ∞] is called F-superharmonic if

1. u is lower semi-continuous;

2. u ≢ ∞ in each component of Ω;

3. u satisfies the comparison principle in each open D ⊂⊂ Ω: If $h\in C(\overline{D})$ is an F-solution in D, and if h ≤ u on ∂D, then h ≤ u in D.

Analagously, a function u : Ω → [−∞, ∞) is called F-subharmonic if

1. u is upper semi-continuous;

2. u ≢ −∞ in each component of Ω;

3. u satisfies the comparsion principle in each open D ⊂⊂ Ω: If $h\in C(\overline{D})$ is an F-solution in D, and if h ≥ u on ∂D, then h ≥ u in D.

We say that u ∈ C(Ω) is F-harmonic if u is both F-subharmonic and F-superharmonic.

Lemma 2.8.

1. If u is F-superharmonic, then au + b is F-superharmonic whenever a and b are real numbers and a ≥ 0.

2. If u and v are F-superhmaronic, then the function min{u, v} is F-superharmonic.

3. Suppose that u_i, i = 1, 2, ⋯, are F-superharmonic in Ω. If the sequence u_i is increasing or converges uniformly on compact subsets of Ω, then in each component of Ω, the limit function u = lim_i→∞ u_i is F-superharmonic unless u ≡ ∞.

Theorem 2.9 (Comparison principle for F-super/subharmonic functions). Suppose that u is F-superharmonic and that v is F-subharmonic in Ω. If

$$\underset{y\to x}{\text{lim sup }}v(y)\le \underset{y\to x}{\text{lim inf }}u(y)$$

for all x ∈ ∂Ω, then v ≤ u in Ω.

Proof. Fix ε > 0 and let

$$K_{\epsilon }≔\left\{x\in \Omega :v(x)\ge u(x)+\epsilon \right\}.$$

Then K_ε is a compact subset of Ω and so there exists an open cover D_ε such that K_ε ⊂ D_ε ⊂ Ω where D_ε is a union of finitely many balls B_i, and ∂D_ε ⊂ Ω\K_ε. Since u is lower semi-continuous, v is upper semi-continuous and ∂D_ε is compact, we can choose a continuous function θ on ∂D_ε such that v ≤ θ ≤ u + ε on ∂D_ε. Moreover, since D_ε satisfies a uniform exterior cone condition, there exists $h\in C(\overline{D})$ which is the unique F-solution in D_ε that coincides with θ on ∂D_ε by applying Theorem 2.6. Now the definition of F-super/subharmonic functions yields that

$$v\le h\le u+\epsilon \text{ in }{D}_{\epsilon }.$$

Hence, v ≤ u + ε in Ω and the desired result follows by letting ε → 0. □

Now we describe the equivalence of F-supersolution and F-superharmonic function; see also [14, 21, 24].

Theorem 2.10. u is an F-supersolution in Ω if and only if u is F-superharmonic in Ω.

Proof. Assume first that u is an F-supersolution in Ω. To show that u is F-superharmonic, we only need to verify the property (iii) in the definition of F-superharmonic functions. Let D ⊂⊂ Ω be an open set and take $h\in C(\overline{D})$ to be an F-solution in D such that h ≤ u on ∂D. Thus, applying the comparison principle for F-super/subsolutions (Theorem 2.5) for u and h, we conclude that h ≤ u in $\overline{D}$.

Assume now that u is F-superharmonic in Ω. For any x₀ ∈ Ω, take φ ∈ C²(B_r(x₀)) such that u − φ has a local minimum 0 at x₀. We need to prove that

$$F\left(D^2\phi \left(x_0\right)\right)\le 0.$$ (2.1)

We argue by contradiction; suppose that (2.1) fails. By the continuity of the operator F, there exist τ > 0 and ρ ∈ (0, r) such that

$$F\left(D^2\phi (x)\right)>\tau \text{ in }{B}_{\rho }\left(x_0\right)\text{.}$$

Consider a cut-off function $\eta \in C_0^2\left(B_{\rho }\left(x_0\right)\right)$ with supp η ⊂ B_ρ/2(x₀) and η(x₀) = 1. Since the uniform ellipticity gives

$$F\left(D^2(\phi +\epsilon \eta )\right)\ge F\left(D^2\phi \right)+\epsilon {𝒫}_{\lambda ,\Lambda }^{-}\left(D^2\eta \right)\text{ for any }\epsilon >0,$$

we can choose a sufficiently small ε₀ > 0 so that

$$F\left(D^2\left(\phi +{\epsilon }_0\eta \right)\right)\ge 0\text{ in }{B}_{\rho }\left(x_0\right).$$

In other words, since φ + ε₀η ∈ C²(B_ρ(x₀)), φ + ε₀η is an F-subsolution in B_ρ(x₀). Furthermore, by a similar argument as in the first part, we have φ + ε₀η is F-subharmonic in B_ρ(x₀). On the other hand, on ∂B_ρ/2(x₀), we have

$$\phi (x)+{\epsilon }_0\eta (x)=\phi (x)\le u(x).$$

Thus, by the comparison principle for F-super/subharmonic functions (Theorem 2.9) for u and φ + ε₀η, we conclude that φ + ε₀η ≤ u in B_ρ/2(x₀). In particular, letting x = x₀, we have φ(x₀) + ε₀ ≤ u(x₀), which contradicts to the fact that u(x₀) = φ(x₀). □

The result for F-subsolution and F-subharmonic function can be derived in the same manner and consequently, a function u is an F-solution if and only if it is F-harmonic.

2.2. Perron’s method.

Lemma 2.11 (Pasting lemma). Let D ⊂ Ω be open. Also let u and v be F-superharmonic in Ω and D, resepctively. If the function

$$s≔\{\begin{array}{ll}\text{min\hspace{0.17em}}\left\{u,v\right\}\hfill & \text{in }D,\hfill \\ u\hfill & \text{in }\Omega \backslash D,\hfill \end{array}$$

is lower semi-continuous, then s is F-superharmonic in Ω.

Proof. Let G ⊂⊂ Ω be open and $h\in C(\overline{G})$ be F-harmonic such that h ≤ s on ∂G. Then h ≤ u in $\overline{G}$. In particular, since s is lower semi-continuous,

$$\underset{D\cap G\ni y\to x}{\text{lim}}h(y)\le u(x)=s(x)\le \underset{D\cap G\ni y\to x}{\text{lim inf }}v(y)$$

for all x ∈ ∂D ∩ G. Thus,

$$\underset{D\cap G\ni y\to x}{\text{lim}}h(y)\le s(x)\le \underset{D\cap G\ni y\to x}{\text{lim inf }}v(y)$$

for all x ∈ ∂(D ∩ G), and Theorem 2.9 implies h ≤ v in D ∩ G. Therefore, h ≤ s in G and the lemma is proved. □

Suppose that u is F-superharmonic in Ω and that B ⊂⊂ Ω is an open ball. Let

$$u_B(z)≔\text{inf\hspace{0.17em}}\left\{v(z):v\text{ is }F\text{-superharmonic in }B,\underset{y\to x}{\text{ lim inf }}v(y)\ge u(x)\text{ for each }x\in \partial B\right\}\text{. }$$

Then define the Poisson modification P(u, B) of u in B to be the function

$$P(u,B)≔\{\begin{array}{ll}{u}_B\hfill & \text{in }B,\hfill \\ u\hfill & \text{in }\Omega \backslash B.\hfill \end{array}$$

Lemma 2.12 (Poisson modification). The Poisson modification P(u, B) is F-superharmonic in Ω, F-harmonic in B, and P(u, B) ≤ u in Ω.

Proof. By definition, it is clear that P(u, B) ≤ u in Ω. To show P(u, B) is F-harmonic in B, choose an increasing sequence of continuous functions {θ_j}_j≥1 on ∂B such that u = lim_j→∞ θ_j. (recall that this is possible since u is lower semi-continuous.) Then let $h_j\in C(\overline{B})$ be the F-solution of the Dirichlet problem F(D²h_j) = 0 in B and h_j = θ_j on ∂B by Theorem 2.6. The comparison principle yields that h_j is also an increasing sequence. Thus, applying Harnack convergence theorem (Theorem 2.4), we have the limit function h = lim_j→∞ h_j is an F-solution in B. Since

$$\underset{y\to x}{\text{lim inf }}h(y)\ge \underset{j\to \infty }{\text{lim}}\underset{y\to x}{\text{lim inf }}{h}_j(y)=\underset{j\to \infty }{\text{lim}}{h}_j(x)=\underset{j\to \infty }{\text{lim}}{\theta }_j(x)=u(x),$$ (2.2)

for any x ∈ ∂B, we have h ≥ P(u, B) in B by the definition of u_B. On the other hand, since h_j(x) ≤ lim inf_y→x v(y) where x ∈ ∂B and v is an admissible function for u_B, we have h ≤ P(u, B) in B by applying the comparison principle, letting j → ∞ and taking the infimum over v. Therefore, P(u, B) = h is F-harmonic in B.

Finally, if we show that P(u, B) is lower semi-continuous, then it immediately follows from the pasting lemma that P(u, B) is F-superharmonic in Ω. Indeed, it is enough to show that P(u, B) is lower semi-continuous at each point x ∈ ∂B; recall (2.2). □

Remark 2.13 (Perron’s method). Let Ω be an open, bounded subset of ${ℝ}^n$ and f be a bounded function on ∂Ω. The upper class 𝒰_f = 𝒰_f(Ω) consists of all functions u in Ω such that

1. u is F-superharmonic in Ω;

2. u is bounded below;

3. lim inf_Ω∋y→x u(y) ≥ f(x) for each x ∈ ∂Ω.

Then, we define the upper Perron solution of f by

$${\overline{H}}_f={\overline{H}}_f(\Omega )≔\underset{u\in {𝒰}_f}{\text{inf\hspace{0.17em}}}u.$$

Similarly, let the lower class ℒ_f = ℒ_f(Ω) be the set of all F-subharmonic functions v in Ω which are bounded above and such that

$$\underset{\Omega \ni y\to x}{\text{lim sup }}v(y)\le f(x)\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}for each }x\in \partial \Omega ,$$

and define the lower Perron solution of f by

$${\underset{\_}{H}}_f={\underset{\_}{H}}_f(\Omega )≔\underset{v\in {ℒ}_f}{\text{sup\hspace{0.17em}}}v.$$

Then, the comparison principle yields that ${\underset{\_}{H}}_f\le {\overline{H}}_f$.

Lemma 2.14. The Perron solutions ${\underset{\_}{H}}_f$ and ${\overline{H}}_f$ are F-solutions in Ω.

Proof. This proof is based on the argument used in [19]. Fix an open ball B with B ⊂⊂ Ω. Next, choose a countable, dense subset X = {x₁, x₂, …} of B and then for each j = 1, 2, …, choose u_i,j ∈ 𝒰_f such that

$$\underset{i\to \infty }{\text{lim}}{u}_{i,j}\left(x_j\right)={\overline{H}}_f\left(x_j\right).$$

Moreover, replacing u_i,j+1 by min{u_i,j, u_i,j+1} if necessary, we have

$$\underset{i\to \infty }{\text{lim}}{u}_{i,j}\left(x_k\right)={\overline{H}}_f\left(x_k\right),$$

for each k = 1, 2…, j and each j. Now, let U_i,j ≔ P(u_i,j, B) be the Poisson modification of u_i,j in B. Then we observe that ${\overline{H}}_f\le U_{i,j}\le u_{i,j}$ and U_i,j is F-harmonic in B. By compactness (Theorem 2.3 (ii)), U_i,j converges locally uniformly to F-harmonic v_j in B (passing to a subsequence, if necessary). Again by compactness, v_j converges locally uniformly to F-harmonic h in B.

By the construction of h, it follows immediately that

$${\overline{H}}_f\le h$$

in B and ${\overline{H}}_f=h$ on X. For any u ∈ 𝒰_f and its Poisson modification U = P(u, B), we have $u\ge U\ge {\overline{H}}_f$. Since U ≥ h on X (which is dense in B) and U, h are continuous in B, it follows that U ≥ h in B. Thus, u ≥ h in B which implies that

$${\overline{H}}_f\ge h$$

in B. Hence, ${\overline{H}}_f=h$ is F-harmonic in Ω and a similar argument for ${\underset{\_}{H}}_f$ completes the proof. □

We emphasize that although we proved that $F\left(D^2{\overline{H}}_f\right)=0$ in Ω, we cannot guarantee that ${\overline{H}}_f$ enjoys the boundary condition of the Dirichlet problem, ${\overline{H}}_f=f$ on ∂Ω. To investigate the boundary behavior of the Perron solutions and ensure the solvability of the Dirichlet problem, we need to introduce further concepts, namely, a regular point and a barrier function.

Definition 2.15 (A regular point). A boundary point x₀ ∈ ∂Ω is (F-)regular with respect to Ω, if

$$\underset{\Omega \ni y\to x_0}{\text{lim}}{\overline{H}}_f(y)=f\left(x_0\right)\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\underset{\Omega \ni y\to x_0}{\text{lim}}{\underset{\_}{H}}_f(y)=f\left(x_0\right)$$

whenever f ∈ C(∂Ω). An open and bounded set Ω is called regular if each x₀ ∈ ∂Ω is a regular boundary point.

Remark 2.16. Suppose that an operator ℳ satisfies ℳ[−u] = −ℳ[u]; for example, any linear operator L and p-Laplcian operators Δ_p possess this property. Then we have

$${\overline{H}}_f=-{\underset{\_}{H}}_{-f},$$

and so in this case, we can equivalently call x₀ ∈ ∂Ω is regular if

$$\underset{\Omega \ni y\to x_0}{\text{lim}}{\overline{H}}_f(y)=f\left(x_0\right)$$

whenever f ∈ C(∂Ω). Nevertheless, for the general fully nonlinear operator F, we do not have this property. Therefore, it seems that we have to require both conditions simultaneously, when we define a regular point for F. To the best of our knowledge, it is unknown whether the two conditions in the definition are redundant. One possible approach to show that only one condition is essential is to prove that f is resolutive whenever f is continuous on ∂Ω; see Definition 2.17 for the definition of resolutivity.

Before we define a barrier function, which characterizes a regular boundary point, we shortly deal with the resolutivity of boundary data:

Definition 2.17 (Resolutivity). We say that a bounded function f on ∂Ω is (F-)resolutive if the upper and the lower Perron solutions ${\overline{H}}_f$ and ${\underset{\_}{H}}_f$ coincide in Ω. When f is resolutive, we write $H_f≔{\overline{H}}_f={\underset{\_}{H}}_f$.

Lemma 2.18. Let Ω be a bounded open set of ${ℝ}^n$, let f and g be bounded functions on ∂Ω, and let c be any real number.

1. If f = c on ∂Ω, then f is resolutive and H_f = c in Ω.

2. ${\overline{H}}_{f+c}={\overline{H}}_f+c$ and ${\underset{\_}{H}}_{f+c}={\underset{\_}{H}}_f+c$. If f is resolutive, then f + c is resolutive and H_f+c = H_f + c.

3. If c > 0, then ${\overline{H}}_{cf}=c{\overline{H}}_f$ and ${\underset{\_}{H}}_{cf}=c{\underset{\_}{H}}_f$ If f is resolutive, then cf is resolutive and H_cf = cH_f for c ≥ 0.

4. If f ≤ g, then ${\overline{H}}_f\le {\overline{H}}_g$ and ${\underset{\_}{H}}_f\le {\underset{\_}{H}}_g$.

Note that the resolutivity of f does not imply

$$\underset{y\to x}{\text{lim\hspace{0.17em}}}{H}_f(y)=f(x)$$

for x ∈ ∂Ω. However, the converse is true in some sense:

Lemma 2.19. Let Ω be an open and bounded subset of ${ℝ}^n$ and f be a bounded function on ∂Ω. Suppose that there exists F-harmonic h in Ω such that

$$\underset{\Omega \ni y\to x}{\text{lim}}h(y)=f(x)$$

for any x ∈ ∂Ω. Then ${\overline{H}}_f=h={\underset{\_}{H}}_f$. In particular, f is resolutive.

Proof. Since h ∈ 𝒰_f ∩ ℒ_f, we have ${\overline{H}}_f\le h\le {\underset{\_}{H}}_f$. □

Lemma 2.20. If u is a bounded F-superharmonic (or F-subharmonic) function on the bounded open set Ω such that f(x) = lim_Ω∋y→x u(y) exists for all x ∈ ∂Ω, then f is a resolutive boundary function.

Proof. Obviously, we have u ∈ 𝒰_f and so ${\overline{H}}_f\le u$ in Ω. Then since ${\overline{H}}_f$ is F-harmonic in Ω and

$$\underset{\Omega \ni y\to x}{\text{lim \hspace{0.17em}sup}}{\overline{H}}_f(y)\le \underset{\Omega \ni y\to x}{\text{lim}}u(y)=f(x),$$

we have ${\overline{H}}_f\in {ℒ}_f$, which implies that ${\overline{H}}_f\le {\underset{\_}{H}}_f$. Because ${\underset{\_}{H}}_f\le {\overline{H}}_f$ always holds, we conclude that f is resolutive. An analogous argument works for the F-subharmonic case. □

2.3. Characterization of a regular point.

Definition 2.21 (Barrier). Let x₀ ∈ ∂Ω. A function $w^{+}:\Omega \to ℝ[\text{resp}.w^{-}]$ is an upper barrier [resp. lower barrier] in Ω at the point x₀ if

1. w⁺ [resp. w⁻] is F-superharmonic [resp. F-subharmonic] in Ω;

2. lim inf_Ω∋y→x w⁺(y) > 0 [resp. lim sup_Ω∋y→x w⁻(y) < 0] for each x ∈ ∂Ω \ {x₀};

3. ${\text{lim}}_{\Omega \ni y\to x_0}{w}^{+}\left(y\right)=0$. $\left[\text{resp}.{\text{ lim}}_{\Omega \ni y\to x_0}{w}^{-}\left(y\right)=0.\right]$

Observe that the maximum principle indicates that an upper barrier w⁺ is positive in Ω and a lower barrier w⁻ is negative in Ω. Moreover, under the condition (F2), cw⁺ is still an upper barrier for any constant c > 0 and an upper barrier w⁺. See also [38].

Now we can deduce that a regular boundary point is characterized by the existence of upper and lower barriers.

Theorem 2.22. Let x₀ ∈ ∂Ω. Then the following are equivalent:

1. x₀ is regular;

2. there exist an upper barrier and a lower barrier at x₀.

Proof. (ii) ⇒ (i) For f ∈ C(∂Ω) and ε > 0, there is δ > 0 such that |x − x₀| ≤ δ with x ∈ ∂Ω implies |f(x)−f(x₀)| < ε. Moreover, for M ≔ sup_∂Ω |f|, there exists a large number K > 0 such that

$$K\cdot \underset{\Omega \ni y\to x}{\text{lim inf }}{w}^{+}(y)\ge 2M\text{ for all }x\in \partial \Omega \text{ with }\left|x-x_0\right|\ge \delta .$$

Here we used that x ↦ lim inf_Ω∋y→x w⁺(y) is lower semi-continuous on ∂Ω. Then since Kw⁺ + f(x₀) + ε ∈ 𝒰_f, we have

$${\overline{H}}_f(y)\le K{w}^{+}(y)+f\left(x_0\right)+\epsilon ,$$

which implies that

$$\underset{\Omega \ni y\to x_0}{\text{lim sup }}{\overline{H}}_f(y)\le f\left(x_0\right).$$

An analogous argument leads to

$$\underset{\Omega \ni y\to x_0}{\text{lim inf }}{\underset{\_}{H}}_f(y)\ge f\left(x_0\right).$$

Since ${\underset{\_}{H}}_f\le {\overline{H}}_f$, we conclude that

$$\underset{\Omega \ni y\to x_0}{\text{lim}}{\overline{H}}_f(y)=f\left(x_0\right)=\underset{\Omega \ni y\to x_0}{\text{lim}}{\underset{\_}{H}}_f(y),$$

i.e. x₀ is a regular boundary point.

(i) ⇒ (ii) Define a distance function d by

$$d(y)≔{\left|y-x_0\right|}^2$$

so that d is continuous, non-negative and d(y) = 0 if and only if y = x₀. Moreover, since D²d = 2I, we have F(D²d) = 2F(I) > 0, i.e. d is F-subharmonic.

Then letting $w^{+}≔{\underset{\_}{H}}_d$, we have w⁺ is F-harmonic in Ω and it follows from d ∈ ℒ_d that w⁺ ≥ d in Ω. Thus, for any x ∈ ∂Ω \ {x₀},

$$\underset{\Omega \ni y\to x}{\text{lim inf }}{w}^{+}(y)\ge d(x)={\left|x-x_0\right|}^2>0.$$

Furthermore, since x₀ is regular, we have

$$\underset{\Omega \ni y\to x_0}{\text{lim}}{w}^{+}(y)=d\left(x_0\right)=0,$$

and so w⁺ is a desired upper barrier. The existence of a lower barrier is guaranteed by considering $\tilde{d}(y)≔-d(y)=-{\left|y-x_0\right|}^2$ and $w^{-}≔{\overline{H}}_{\tilde{d}}$. □

Indeed, the barrier characterization is a local property:

Lemma 2.23. Let x₀ ∈ ∂Ω and G ⊂ Ω be open with x₀ ∈ ∂G. If x₀ is regular with respect to Ω, then x₀ is regular with respect to G.

Proof. By Theorem 2.22, there exist an upper barrier w⁺ and a lower barrier w⁻ with respect to Ω at x₀. Then w⁺|_G and w⁻|_G become the desired barriers with respect to G at x₀. Again by Theorem 2.22, x₀ is regular with respect to G. □

Lemma 2.24. Let x₀ ∈ ∂Ω and B be a ball containing x₀. Then x₀ is regular with respect to Ω if and only if x₀ is regular with respect to B ∩ Ω.

Proof. By Lemma 2.23, one direction is immediate. For the opposite direction, suppose that x₀ is regular with respect to B ∩ Ω. Then there exist an upper barrier w⁺ and a lower barrier w⁻ with respect to B ∩ Ω. If we let m := min_∂B∩Ω w⁺ > 0 (the minimum exists because w⁺ is lower semi-continuous), then the pasting lemma, Lemma 2.11, shows that

$$s^{+}≔\{\begin{array}{ll}\text{min\hspace{0.17em}}\left\{w^{+},m\right\}\hfill & \text{in }B\cap \Omega ,\hfill \\ m\hfill & \text{in }\Omega \backslash B,\hfill \end{array}$$

is F-superharmonic in Ω. One can easily verify that s⁺ is an upper barrier with respect to Ω at x₀. Similarly, a lower barrier s⁻ can be constructed. □

The barrier characterization leads to another useful corollary, which enables us to write x₀ is regular instead of F-regular, without ambiguity.

Corollary 2.25. A boundary point x₀ ∈ ∂Ω is F-regular if and only if x₀ is $\tilde{F}$-regular.

Proof. Suppose that x₀ is F-regular. By Theorem 2.22, there exists an upper barrier $w_F^{+}$ and a lower barrier $w_F^{-}$. If we let $w_{\tilde{F}}^{+}≔-w_F^{-}$ and $w_{\tilde{F}}^{-}≔-w_F^{+}$, then $w_{\tilde{F}}^{+}$ and $w_{\tilde{F}}^{-}$ become an upper barrier and a lower barrier for $\tilde{F}$, respectively. Therefore, again by Theorem 2.22, x₀ is $\tilde{F}$-regular. □

Now we present one sufficient condition that guarantees a regular boundary point, namely the exterior cone condition. In Section 4, we suggest another sufficient condition, namely the Wiener criterion, which contains this exterior cone condition as a special case.

Theorem 2.26 (Exterior cone condition). Suppose that Ω satisfies an exterior cone condition at x₀ ∈ ∂Ω. Then x₀ is a regular boundary point.

Proof. The proof relies on the construction of a local barrier at x₀. See [9, 34, 37] for details. □

Corollary 2.27. All polyhedra and all balls are regular. Furthermore, every open set can be exhausted by regular open sets. Here a bounded open set Ω is called a polyhedron if $\partial \Omega =\partial \overline{\Omega }$ and if ∂Ω is contained in a finite union of (n − 1)-hyperplanes.

Proof. Since polyhedra and balls satisfy the uniform exterior cone condition, the first assertion follows from Theorem 2.26. For the second assertion, exhaust Ω by domains D₁ ⊂⊂ D₂ ⊂⊂ ⋯ ⊂⊂ Ω. Then, since $\overline{D_j}$ is compact, there exists a finite union of open cubes Q_ji(⊂ D_j+1) that covers $\overline{D_j}$. Letting $P_j≔{\cup }_i\text{ int }\overline{Q_{j_i}}$ which is a polyhedron by the construction, we obtain the desired exhaustion. □

3. Balayage and capacity

3.1. Balayage and capacity potential.

We define the lower semi-continuous regularization $\widehat{u}$ of any function u : E → [−∞, ∞] by

$$\widehat{u}(x)≔\underset{r\to 0}{\text{lim}}\underset{E\cap B_r(x)}{\text{inf\hspace{0.17em}}}u.$$

Lemma 3.1. Suppose that ℱ is a family of F-superharmonic functions in Ω, locally uniformly bounded below. Then the lower semi-continuous regularization s of inf ℱ,

$$s(x)=\underset{r\to 0}{\text{lim}}\underset{B_r(x)}{\text{inf\hspace{0.17em}}}(\text{inf\hspace{0.17em}}ℱ),$$

is F-superharmonic in Ω.

Proof. Since ℱ is locally uniformly bounded below, s is lower semi-continuous. Fix an open D ⊂⊂ Ω and let $h\in C(\overline{D})$ be an F-harmonic function satisfying h ≤ s on ∂D. Then h ≤ u in D whenever u ∈ ℱ. It follows from the continuity of h that h ≤ s in D. □

Definition 3.2 (Balayage and capacity potential).

1. For ψ : Ω → (−∞, ∞] which is locally bounded below, let

   $${\Phi }^{\psi }={\Phi }^{\psi }(\Omega )≔\left\{u:u\text{ is }F\text{-superharmonic in }\Omega \text{ and }u\ge \psi \text{ in }\Omega \right\}.$$
   Then the function

   $$R^{\psi }=R^{\psi }(\Omega )≔\text{inf\hspace{0.17em}}{\Phi }^{\psi }$$

   is called the reduced function and its lower semi-continuous regularization

   $${\widehat{R}}^{\psi }={\widehat{R}}^{\psi }(\Omega )$$

   is called the balayage of ψ in Ω. By Lemma 3.1, ${\widehat{R}}^{\psi }$ is F-superharmonic in Ω.
2. If u is a non-negative function on a set E ⊂ Ω, we write

   $${\Phi }_E^u={\Phi }^{\psi },R_E^u=R^{\psi },{\widehat{R}}_E^u={\widehat{R}}^{\psi },$$

   where

   $$\psi =\{\begin{array}{ll}u\hfill & \text{in }E,\hfill \\ 0\hfill & \text{in }\Omega \backslash E.\hfill \end{array}$$

   The function ${\widehat{R}}_E^u$ is called the balayage of u relative to E.

3. In particular, we call the function ${\widehat{R}}_E^1$ the (F-)capacity potential of E in Ω.

Remark 3.3. For an operator in divergence form, there exists an alternative method to define the capacity potential. For simplicity, suppose that the operator is given by the p-Laplacian. Let Ω be bounded and K ⊂ Ω be a compact set. For $\psi \in C_0^{\infty }(\Omega )$ with ψ ≡ 1 on K, the p-harmonic function u in Ω \ K with $u-\psi \in W_0^{1,p}(\Omega \backslash K)$ is called the capacity potential of K in Ω and denoted by ℛ(K, Ω). Here note that ℛ(K, Ω) is independent of the particular choice of ψ and the existence of the capacity potential is guaranteed by the variational method. Indeed, both definitions of capacity potentials coincide; see [12, Chapter 9] for details.

Lemma 3.4. The balayage ${\widehat{R}}_E^u$ is F-harmonic in $\Omega \backslash \overline{E}$ and coincides with ${\widehat{R}}_E^u$ there. If, in addition, u is F-superharmonic in Ω, then ${\widehat{R}}_E^u=u$ in the interior of E.

Proof. Observe first that if v₁ and v₂ are in ${\Phi }_E^u$, then so is min{v₁, v₂}. Hence, the family ${\Phi }_E^u$ is downward directed and we may invoke Choquet’s topological lemma (see Lemma 8.3. in [12]): there is a decreasing sequence of functions $v_i\in {\Phi }_E^u$ with the limit v such that

$$\widehat{v}(x)={\widehat{R}}_E^u(x)$$

for all x ∈ Ω.

Next, we choose a ball $B\subset \subset \Omega \backslash \overline{E}$ and consider a Poisson modification s_i = P(v_i, B). Then it follows that $s_i\in {\Phi }_E^u$ and s_i+1 ≤ s_i ≤ v_i. Thus, we have

$$R_E^u\le s≔\underset{i\to \infty }{\text{lim}}{s}_i\le v,$$

which implies that ${\widehat{R}}_E^u=\widehat{v}=\widehat{s}$. Moreover, since s is F-harmonic in B (Harnack convergence theorem, Theorem 2.4), we know that $\widehat{s}=s$. Therefore, we conclude that the balayage ${\widehat{R}}_E^u$ is F-harmonic in $\Omega \backslash \overline{E}$. The second assertion of the lemma is rather immediate since $u\in {\Phi }_E^u$ if u is F-superharmonic in Ω.

Lemma 3.5. Let K be a compact subset of Ω and consider $R_K^1=R_K^1(\Omega )$ and ${\widehat{R}}_K^1={\widehat{R}}_K^1(\Omega )$.

1. $0\le {\widehat{R}}_K^1\le R_K^1\le 1$ in Ω.

2. $R_K^1=1$ in K.

3. $R_K^1={\widehat{R}}_K^1$ in (∂K)^c.

4. ${\widehat{R}}_K^1$ is F-superharmonic in Ω and F-harmonic in Ω \ K.

Proof.

1. It immediately follows from the definition of $R_K^1$ and the comparison principle.

2. Since 1 ∈ Φ^ψ(Ω), we have $R_K^1\le 1$ in Ω. On the other hand, for any u ∈ Φ^ψ(Ω), we have u ≥ ψ = 1 in K and so $R_K^1\ge 1$ in K.

3. (iii), (iv) It immediately follows from Lemma 3.4 and part (ii). □

The following theorem shows that the capacity potential can be understood as the upper Perron solution:

Theorem 3.6. Suppose that K is a compact subset of a bounded, open set Ω and that $u={\widehat{R}}_K^1(\Omega )$ is the capacity potential of K in Ω. Moreover, let f be a function such that

$$f=\{\begin{array}{ll}1\hfill & on\partial K,\hfill \\ 0\hfill & in\partial \Omega .\hfill \end{array}$$

Then

$${\widehat{R}}_K^1(\Omega )={\overline{H}}_f(\Omega \backslash K)$$

in Ω \ K.

Proof. Lemma 3.4 shows that ${\widehat{R}}_K^1=R_K^1$ in Ω \ K. Then recall that

$$R_K^1(\Omega )=\text{inf }{\Phi }_K^1=\text{inf\hspace{0.17em}}\left\{v:v\text{ is }F\text{-superharmonic in }\Omega \text{ and }v\ge \psi \text{ in }\Omega \right\},$$

where

$$\psi =\{\begin{array}{ll}1\hfill & \text{in }K,\hfill \\ 0\hfill & \text{in }\Omega \backslash K,\hfill \end{array}$$

and

$$\begin{array}{l}{\overline{H}}_f(\Omega \backslash K)=\text{inf }{𝒰}_f\hfill \\ =\text{inf\hspace{0.17em}}\left\{v:v\text{ is }F\text{-superharmonic in }\Omega \backslash K,\underset{\Omega \backslash K\ni y\to x}{\text{ lim\hspace{0.17em}inf}}v(y)\ge f(x)\text{ for each }x\in \partial (\Omega \backslash K)\right\},\hfill \end{array}$$

where

$$f=\{\begin{array}{ll}1\hfill & \text{on }\partial K,\hfill \\ 0\hfill & \text{in }\partial \Omega .\hfill \end{array}$$

1. Suppose that $v\in {\Phi }_K^1$. Since v ≥ 0 in Ω, we have lim inf_{Ω\K∋y→x} v(y) ≥ 0 = f(x) for x ∈ ∂Ω. Moreover, since v is lower semi-continuous, we have

   $$\underset{\Omega \backslash K\ni y\to x}{\text{lim inf }}v(y)\ge \underset{\Omega \ni y\to x}{\text{lim inf }}v(y)\ge v(x)\ge 1=f(x),$$

   for x ∈ ∂K. Therefore, we conclude v ∈ 𝒰_f, which implies that ${\overline{H}}_f(\Omega \backslash K)\le R_K^1(\Omega )$ in Ω \ K.
2. Suppose that v ∈ 𝒰_f. We consider $\overline{v}≔\text{min\hspace{0.17em}}\left\{1,v\right\}\in {𝒰}_f$ so that $0\le \overline{v}\le 1$ in Ω \ K. Then, since u ≡ 1 is F-superharmonic in Ω, the function

   $$s=\{\begin{array}{ll}\text{min\hspace{0.17em}}\left\{1,\overline{v}\right\}=\overline{v}\hfill & \text{in }\Omega \backslash K,\hfill \\ 1\hfill & \text{in }K\hfill \end{array}$$

   is F-superharmonic in Ω by pasting lemma, Lemma 2.11. Obviously, $s\in {\Phi }_K^1$ and so $R_K^1(\Omega )\le \overline{v}\le v$ in Ω \ K. □

3.2. Capacity.

In general, for an operator in divergence form, we consider a variational capacity, which comes from minimizing the energy among admissible functions. On the other hand, for an operator in non-divergence form, we cannot consider the corresponding energy, and so we require an alternative approach to attain a proper notion of capacity. Our definition of a capacity is in the same context with Bauman [4] (for linear operators in non-divergence form) and Labutin [24] (for the Pucci extremal operators).

Definition 3.7 (Non-variational capacity). For a ball B = B_2r(x₀), we fix a ball B′ = B_7/5r(x₀) ⊂ B and a point $y_0=x_0+\frac{3}{2}r{e}_1$. Then we define a capacity for fully nonlinear operator F by cap(K, B) = cap_F (K, B) ≔ inf{u(y₀) : u is F-superharmonic in B, u ≥ 0 in B, and u ≥ 1 in K} whenever K is a compact subset of B′.

Comparing the definitions of capacity and capacity potential, we immediately notice that

$$\text{cap}(K,B)={\widehat{R}}_K^1(B)\left(y_0\right).$$

Moreover, appealing to Theorem 3.6, we further have

$$\text{cap}(K,B)={\overline{H}}_f(B\backslash K)\left(y_0\right),$$

where the boundary data f on ∂(B \ K) is given by

$$f=\{\begin{array}{ll}1\hfill & \text{on }\partial K,\hfill \\ 0\hfill & \text{in }\partial B.\hfill \end{array}$$

Finally, considering Harnack inequality for ${\widehat{R}}_K^1(B)$ on the sphere ∂B_3r/2(x₀), we notice that capacities defined for different choices of y₀ ∈ ∂B_3r/2(x₀) are comparable.

Lemma 3.8 (Properties of capacity). Fix a ball B = B_2r(x₀). Then the set function K ↦ cap(K, B), where K is a compact subset of B′ = B_7/5r(x₀), enjoys the following properties:

1. 0 ≤ cap(K, B) ≤ 1.

2. If K₁ ⊂ K₂ ⊂ B′, then

   $$\text{cap}\left(K_1,B\right)\le \text{cap}\left(K_2,B\right).$$
3. If a monotone sequence of compact sets ${\left\{K_j\right\}}_{j=1}^{\infty }$ satisfies B′ ⊃ K₁ ⊃ K₂ ⊃ ⋯, then

   $$\text{cap}(K,B)=\underset{j\to \infty }{\text{lim}}\text{cap}\left(K_j,B\right),\mathit{\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}for }}K≔\underset{j=1}{\overset{\infty }{\cap }}{K}_j\text{.}$$
4. (Subadditivity) We further suppose that F is convex. If K₁ and K₂ are compact subsets of B′, then

   $$\text{cap}\left(K_1\cup K_2,B\right)\le \text{cap}\left(K_1,B\right)+\text{cap}\left(K_2,B\right).$$

Proof.

1. Recalling Lemma 3.5, we have 0 ≤ cap(K, B) ≤ 1.

2. If K₁ ⊂ K₂, then ${\Phi }_{K_2}^1\subset {\Phi }_{K_1}^1$ and so cap(K₁, B) ≤ cap(K₂, B).

3. Since cap(K_j, B) ≥ cap(K, B) by (ii), it is immediate that

   $$\text{cap}(K,B)\le \underset{j\to \infty }{\text{lim}}\text{cap}\left(K_j,B\right).$$
   For the reversed inequality, fix small ε > 0 and $u\in {\Phi }_K^1(B)$. If j is large enough, then K_j ⊂ {u ≥ 1 − ε} and so

   $$\underset{j\to \infty }{\text{lim}}\text{cap}\left(K_j,B\right)\le \text{cap}(\left\{u\ge 1-\epsilon \right\},B)\le \frac{1}{1-\epsilon }u\left(y_0\right).$$
   Letting ε → 0⁺ and taking infimum for $u\in {\Phi }_K^1(B)$, we conclude that

   $$\underset{j\to \infty }{\text{lim}}\text{cap}\left(K_j,B\right)\le \text{cap}(K,B).$$
4. Let $v_1\in {\Phi }_{K_1}^1(B)$ and $v_2\in {\Phi }_{K_2}^1(B)$. Since F is convex, we can apply [6, Theorem 5.8] to obtain $\frac{1}{2}\left(v_1+v_2\right)$ is F-superharmonic in B. Moreover, it follows from the assumption (F2) that $v_1+v_2\in {\Phi }_{K_1\cup K_2}^1(B)$ and so $R_{K_1\cup K_2}^1(B)\le v_1+v_2$. Putting the infimum on this inequality and evaluating at y₀, we conclude that

   $$\text{cap}\left(K_1\cup K_2,B\right)\le \text{cap}\left(K_1,B\right)+\text{cap}\left(K_2,B\right).$$

□

We would like to remove the restriction of compact sets when defining a capacity. For this purpose, when U ⊂ B′ is open, we set the inner capacity

$${\text{cap}}_{*}(U,B)≔\underset{K\subset U,K\text{ compact}}{\text{sup\hspace{0.17em}}}\text{cap}(K,B).$$

Then for an arbitrary set E ⊂ B′, we set the outer capacity

$${\text{cap}}^{*}(E,B)≔\underset{E\subset U\subset B^{\prime },U\text{ open}}{\text{inf\hspace{0.17em}}}{\text{cap}}_{*}(U,B).$$

Lemma 3.9. Fix a ball B = B_2r(x₀). For a compact subset K of B′ = B_7r/5(x₀), we have

$$\text{cap}(K,B)={\text{cap}}^{*}(K,B).$$

In other words, there is no ambiguity in having two different definitions for the capacity of compact sets.

Proof.

1. For any open set U satisfying K ⊂ U ⊂ B′, the definition of the inner capacity yields that

   $$\text{cap}(K,B)\le {\text{cap}}_{*}(U,B).$$
   By taking the infimum over such U, we conclude that

   $$\text{cap}(K,B)\le {\text{cap}}^{*}(K,B).$$
2. Define a sequence of compact sets ${\left\{K_j\right\}}_{j=1}^{\infty }$ by

   $$K_j≔\left\{x\in {ℝ}^n:\text{dist}(x,K)\le 1/j\right\},$$

   and a sequence of open sets ${\left\{U_j\right\}}_{j=1}^{\infty }$ by

   $$U_j≔\left\{x\in {ℝ}^n:\text{dist}(x,K)<1/j\right\}.$$
   We may assume K₁ ⊂ B′. Then we have

   $$B^{\prime }\supset K_1\supset U_1\supset K_2\supset U_2\supset \cdots \supset K,\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}K=\underset{j}{\cap }{K}_j\text{. }$$
   Applying Lemma 3.8 (ii), it follows that

   $${\text{cap}}_{*}\left(U_j,B\right)\le \text{cap}\left(K_j,B\right).$$
   By the definition of outer capacity,

   $${\text{cap}}^{*}(K,B)\le {\text{cap}}_{*}\left(U_j,B\right)\le \text{cap}\left(K_j,B\right),\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}for any }j\in \mathbb{N}.$$
   Now letting j → ∞, Lemma 3.8 (iii) leads to

   $${\text{cap}}^{*}(K,B)\le \text{cap}(K,B).$$

□

Roughly speaking, we have the following correspondance:

$$\begin{array}{l}\text{the variational capacity }\leftrightarrow \text{ divergence operator,}\hfill \\ \text{the height capacity }\leftrightarrow \text{ non-divergence operator.}\hfill \end{array}$$

In the following lemma, we explain why the definition of height capacity is reasonable in some sense. In other words, we claim that for the Laplacian operator Δ, two definitions of capacity are comparable.

Lemma 3.10 (The variational capacity and the height capacity). Suppose n ≥ 3 and fix two balls B = B_2r(x₀), B′ = B_7r/5(x₀) and a point $y_0=\frac{3}{2}r{e}_1+x_0\in \partial B_{3r/2}\left(x_0\right)$. Then for any compact set K ⊂ B′, we have

$${\text{cap}}_{\Delta ,\text{var}}(K,B)~{\text{cap}}_{\Delta ,\text{height}}(K,B)r^{n-2},$$

where the comparable constant depends only on n.

Proof. We may assume x₀ = 0. We denote by u the capacity potential with respect to K in B.

Note that u is harmonic in B \ K.

We begin with the variational capacity:

$${\text{cap}}_{\Delta ,\text{var}}(K,B)={\int }_{B\backslash K}|\nabla u{|}^2 \text{d}x={\int }_{\partial K}\frac{\partial u}{\partial n}\text{d}s=-{\int }_{\partial B}\frac{\partial u}{\partial n}\text{d}s.$$

Here we applied the divergence theorem and used the behavior of u on the boundary.

On the other hand, recalling the definition of height capacity, we have

$${\text{cap}}_{\Delta ,\text{height}}(K,B)=u\left(y_0\right).$$

By Harnack inequality, there exist constants c₁, c₂ > 0 which only depend on n such that

$$c_{1}u\left(y_0\right)\le u(x)\le c_{2}u\left(y_0\right)\text{ for any }x\in \partial B_{3r/2}.$$

Thus, if we set m₋ := min_∂B3r/2 u and m₊ := max_∂B3r/2 u, then we have

$$c_1{\text{cap}}_{\Delta ,\text{height}}(K,B)\le m_{-}\le m_{+}\le c_2{\text{cap}}_{\Delta ,\text{height}}(K,B).$$

Moreover, we consider two barriers h^± which solve the Dirichlet problem in B_2r \ B_3r/2:

$$\{\begin{array}{ll}\Delta h^{\pm }=0\hfill & \text{in }{B}_{2r}\backslash B_{3r/2}\hfill \\ h^{\pm }=m_{\pm }\hfill & \text{on }\partial B_{3r/2},\hfill \\ h^{\pm }=0\hfill & \text{on }\partial B_{2r}.\hfill \end{array}$$

Indeed, using the homogeneous solution V (x) = |x|²⁻ⁿ, one can compute h^± explicitly:

$$h^{\pm }(x)=m_{\pm }\cdot \frac{|x{|}^{2-n}-{(2r)}^{2-n}}{{(3r/2)}^{2-n}-{(2r)}^{2-n}}.$$

Then the comparison principle between u and h^± leads to

$$h^{-}\le u\le h^{+}\text{ in }{B}_{2r}\backslash B_{3r/2},$$

and so

$$\frac{c(n)m_{-}}{r}=-\frac{\partial h^{-}}{\partial n}\le -\frac{\partial u}{\partial n}\le -\frac{\partial h^{+}}{\partial n}=\frac{c(n)m_{+}}{r}\text{ on }\partial B.$$

Therefore, we conclude that

$$c_1(n)r^{n-2}{\text{cap}}_{\Delta ,\text{height}}(K,B)\le {\text{cap}}_{\Delta ,\text{var}}(K,B)\le c_2(n)r^{n-2}{\text{cap}}_{\Delta ,\text{height}}(K,B).$$

□

Next, we estimate the capacity of a ball B_ρ with respect to the larger ball B_2r. Indeed, the capacity of a ball can capture the growth rate of the homogeneous solution V of F.

Lemma 3.11 (Capacitary estimate for balls). Let B = B_2r(x₀), $B^{\prime }=B_{\frac{7}{5}r}\left(x_0\right)$ and $y_0=x_0+\frac{3}{2}r{e}_1$.

Then for any $0<\rho <\frac{7}{5}r$, there exists a constant c = c(n, λ, Λ) > 0 which is independent of r and ρ such that

1. (α* > 0)

   $$\frac{1}{c}\frac{r^{-{\alpha }^{*}}}{{\rho }^{-{\alpha }^{*}}-{(2r)}^{-{\alpha }^{*}}}\le {\text{cap}}_F\left(\overline{B_{\rho }\left(x_0\right)},B_{2r}\left(x_0\right)\right)\le \frac{c{r}^{-{\alpha }^{*}}}{{\rho }^{-{\alpha }^{*}}-{(2r)}^{-{\alpha }^{*}}}.$$
2. (α* < 0)

   $$\frac{1}{c}\frac{r^{-{\alpha }^{*}}}{{(2r)}^{-{\alpha }^{*}}-{\rho }^{-{\alpha }^{*}}}\le {\text{cap}}_F\left(\overline{B_{\rho }\left(x_0\right)},B_{2r}\left(x_0\right)\right)\le \frac{c{r}^{-{\alpha }^{*}}}{{(2r)}^{-{\alpha }^{*}}-{\rho }^{-{\alpha }^{*}}}.$$
3. (α* = 0)

   $$\frac{1}{c}\frac{1}{\text{log}(2r)-\text{log }\rho }\le {\text{cap}}_F\left(\overline{B_{\rho }\left(x_0\right)},B_{2r}\left(x_0\right)\right)\le \frac{c}{\text{log}(2r)-\text{log }\rho }.$$

Proof. We may assume x₀ = 0. Applying the argument after the definition of a capacity, we have

$${\text{cap}}_F\left(\overline{B_{\rho }},B_{2r}\right)={\widehat{R}}_{\overline{B_{\rho }}}\left(B_{2r}\right)\left(y_0\right)={\overline{H}}_f\left(B_{2r}\backslash \overline{B_{\rho }}\right)\left(y_0\right),$$

where the boundary data f is given by

$$f=\{\begin{array}{ll}1\hfill & \text{on }\partial B_{\rho },\hfill \\ 0\hfill & \text{in }\partial B_{2r}.\hfill \end{array}$$

Moreover, since a ball is a regular domain, we can write ${\overline{H}}_f\left(B_{2r}\backslash \overline{B_{\rho }}\right)=v$ where v is the unique solution of the Dirichlet problem

$$\{\begin{array}{ll}F\left(D^{2}v\right)=0\hfill & \text{in }{B}_{2r}\backslash \overline{B_{\rho }},\hfill \\ v=1\hfill & \text{on }\partial B_{\rho },\hfill \\ v=0\hfill & \text{in }\partial B_{2r}.\hfill \end{array}$$

Note that ${\overline{H}}_f\left(B_{2r}\backslash \overline{B_{\rho }}\right)$ is continuous upto the boundary. We now split three cases according to the sign of α*(F).

1. (α* > 0) In this case, for the homogeneous solution $V(x)=|x{|}^{-{\alpha }^{*}}V\left(\frac{x}{\left|x\right|}\right)$, denote

   $$V_{+}≔\underset{\left|x\right|=1}{\text{max\hspace{0.17em}}}V(x)\text{ and }{V}_{-}≔\underset{\left|x\right|=1}{\text{min\hspace{0.17em}}}V(x)$$

   and choose two points x₊, x₋ with |x₊| = 1 = |x₋| so that

   $$V\left(x_{+}\right)=V_{+}\text{ and }V\left(x_{-}\right)=V_{-}.$$
   We define two functions

   $$v^{+}(x)≔\frac{V(x)-{(2r)}^{-{\alpha }^{*}}{V}_{-}}{\left[{\rho }^{-{\alpha }^{*}}-{(2r)}^{-{\alpha }^{*}}\right]V_{-}}\text{ and }{v}^{-}(x)≔\frac{V(x)-{(2r)}^{-{\alpha }^{*}}{V}_{+}}{\left[{\rho }^{-{\alpha }^{*}}-{(2r)}^{-{\alpha }^{*}}\right]V_{+}}.$$
   Then we have

   $$F\left(D^2{v}^{+}\right)=0=F\left(D^2{v}^{-}\right)\text{ in }{B}_{2r}\backslash \overline{B_{\rho }},$$

   $$v^{+}\ge 1\text{ on }\partial B_{\rho }\text{ and }{v}^{+}\ge 0\text{ on }\partial B_{2r},$$

   $$v^{-}\le 1\text{ on }\partial B_{\rho }\text{ and }{v}^{-}\le 0\text{ on }\partial B_{2r}.$$
   Thus, the comparison principle yields that

   $$v^{-}\le v={\overline{H}}_f\left(B_{2r}\backslash \overline{B_{\rho }}\right)={\widehat{R}}_{\overline{B_{\rho }}}\left(B_{2r}\right)\le v^{+}\text{ in }{B}_{2r}\backslash \overline{B_{\rho }}.$$
   Finally, applying Harnack inequality for v on ∂B_3r/2, there exists a constant c₁ > 0 which is independent of r > 0 such that

   $$\frac{1}{c_1}v\left(\frac{3r{x}_{+}}{2}\right)\le v\left(y_0\right)\le c_{1}v\left(\frac{3r{x}_{-}}{2}\right).$$
   Therefore, we have the desired upper bound:

   $$\begin{gathered} {\text{cap}}_F\left(\overline{B_{\rho }},B_{2r}\right)=v\left(y_0\right)\le c_{1}v\left(\frac{3r{x}_{-}}{2}\right)\le c_1{v}^{+}\left(\frac{3r{x}_{-}}{2}\right)=c_1\frac{{(3r/2)}^{-{\alpha }^{*}}-{(2r)}^{-{\alpha }^{*}}}{{\rho }^{-{\alpha }^{*}}-{(2r)}^{-{\alpha }^{*}}} \\ =\frac{c{r}^{-{\alpha }^{*}}}{{\rho }^{-{\alpha }^{*}}-{(2r)}^{-{\alpha }^{*}}}. \end{gathered}$$
   Similarly, we derive the lower bound:

   $$\begin{gathered} {\text{cap}}_F\left(\overline{B_{\rho }},B_{2r}\right)=v\left(y_0\right)\ge \frac{1}{c_1}v\left(\frac{3r{x}_{+}}{2}\right)\ge \frac{1}{c_1}{v}^{-}\left(\frac{3r{x}_{+}}{2}\right)=\frac{1}{c_1}\frac{{(3r/2)}^{-{\alpha }^{*}}-{(2r)}^{-{\alpha }^{*}}}{{\rho }^{-{\alpha }^{*}}-{(2r)}^{-{\alpha }^{*}}} \\ =\frac{1}{c}\frac{r^{-{\alpha }^{*}}}{{\rho }^{-{\alpha }^{*}}-{(2r)}^{-{\alpha }^{*}}}. \end{gathered}$$
2. (α* < 0) For simplicity, we assume that the upward-pointing homogeneous solution is given by

   $$V(x)=-|x{|}^{-{\alpha }^{*}}.$$
   Then we can explicitly write the capacity potential:

   $$v(x)=\frac{{(2r)}^{-{\alpha }^{*}}-|x{|}^{-{\alpha }^{*}}}{{(2r)}^{-{\alpha }^{*}}-{\rho }^{-{\alpha }^{*}}}.$$
   Thus,

   $${\text{cap}}_F\left(\overline{B_{\rho }},B_{2r}\right)=v\left(y_0\right)~\frac{r^{-{\alpha }^{*}}}{{(2r)}^{-{\alpha }^{*}}-{\rho }^{-{\alpha }^{*}}}.$$
   For general V, we can compute by a similar argument as in part (i). For example, if $V\left(x\right)=-|x{|}^{-{\alpha }^{*}}V\left(\frac{x}{\left|x\right|}\right)$, then define

   $$v^{+}(x)≔\frac{{(2r)}^{-{\alpha }^{*}}{V}_{+}+V(x)}{\left[{(2r)}^{-{\alpha }^{*}}-{\rho }^{-{\alpha }^{*}}\right]V_{+}}\text{ and }{v}^{-}(x)≔\frac{{(2r)}^{-{\alpha }^{*}}{V}_{-}+V(x)}{\left[{(2r)}^{-{\alpha }^{*}}-{\rho }^{-{\alpha }^{*}}\right]V_{-}}.$$
3. (α* = 0) Again for simplicity, we may assume the upward-pointing homogeneous solution is given by

   $$V(x)=-\text{log}\left|x\right|.$$
   Similarly, we can explicitly write the capacity potential:

   $$v(x)=\frac{\text{log}(2r)-\text{log}\left|x\right|}{\text{log}(2r)-\text{log }\rho }.$$
   Thus,

   $${\text{cap}}_F\left(\overline{B_{\rho }},B_{2r}\right)=v\left(y_0\right)~\frac{1}{\text{log}(2r)-\text{log }\rho }.$$
   For general V, we can compute by a similar argument as in part (i). For example, if $V\left(x\right)=V\left(\frac{x}{\left|x\right|}\right)-\text{log}\left|x\right|$, then define

   $$v^{+}(x)≔\frac{\text{log}(2r)-V_{-}+V(x)}{\text{log}(2r)-\text{log }\rho }\text{ and }{v}^{-}(x)≔\frac{\text{log}(2r)-V_{+}+V(x)}{\text{log}(2r)-\text{log }\rho }.$$

□

We can observe that the capacity of a single point is determined according to the sign of the scaling exponent α*(F). In fact, one can expect the results of the following lemma taking ρ → 0⁺ in the capacitary estimate, Lemma 3.11.

Lemma 3.12. For $z_0\in {ℝ}^n$, choose a ball B = B_2r(x₀) so that z₀ ∈ B′ = B_7r/5(x₀).

1. If α*(F) ≥ 0, then

   $${\text{cap}}_F\left(\left\{z_0\right\},B\right)=0.$$
2. If α*(F) < 0, then

   $${\text{cap}}_F\left(\left\{z_0\right\},B\right)>0.$$

Proof.

1. Let

   $$V(x)=\{\begin{array}{ll}|x{|}^{-{\alpha }^{*}}V\left(\frac{x}{\left|x\right|}\right)\hfill & \text{if }{\alpha }^{*}>0,\hfill \\ -\text{log}\left|x\right|+V\left(\frac{x}{\left|x\right|}\right)\hfill & \text{if }{\alpha }^{*}=0.\hfill \end{array}$$

   be the homogeneous solution of F. Then for m ≔ min_x∈∂B V (x − z₀) and any ε > 0, we have

   $$\epsilon \cdot \left[V\left(x-z_0\right)-m\right]\in {\Phi }_{\left\{z_0\right\}}^1$$

   due to the minimum principle and ${\text{lim}}_{x\to z_0}V\left(x-z_0\right)=\infty$. Thus,

   $$\text{cap}\left(\left\{z_0\right\},B\right)={\widehat{R}}_{\left\{z_0\right\}}^1\left(y_0\right)=R_{\left\{z_0\right\}}^1\left(y_0\right)\le \epsilon \cdot \left[V\left(y_0-z_0\right)-m\right].$$

   Since ε > 0 is arbitrary, we finish the first part of proof.

2. Let $V(x)=-|x{|}^{-{\alpha }^{*}}V\left(\frac{x}{\left|x\right|}\right)$ be the homogeneous solution of F. Then for max_x∈∂B V (x − z₀) ≕ −M < 0, we consider

   $$u(x)≔1+\frac{V\left(x-z_0\right)}{M}.$$
   Since sup_∂B u = 0 and V is a homogeneous function, we have ${\text{sup\hspace{0.17em}}}_{\partial B_{7/5r}}u>0$. On the other hand, recalling Theorem 3.6,

   $${\widehat{R}}_{\left\{z_0\right\}}^1={\overline{H}}_f\left(\Omega \backslash \left\{z_0\right\}\right)\ge {\underset{\_}{H}}_f\left(\Omega \backslash \left\{z_0\right\}\right),$$

   where the boundary data f is given by

   $$f(x)=\{\begin{array}{ll}1\hfill & \text{if }x=z_0,\hfill \\ 0\hfill & \text{if }x\in \partial B.\hfill \end{array}$$
   Then u ∈ ℒ_f and so ${\underset{\_}{H}}_f\left(\Omega \backslash \left\{z_0\right\}\right)\ge u$. Therefore, we conclude that

   $$\underset{\partial B_{7/5r}}{\text{sup\hspace{0.17em}}}{\widehat{R}}_{\left\{z_0\right\}}^1>0$$

   and by Harnack inequality, cap({z₀}, B) > 0 as desired.

□

3.3. Capacity zero sets.

Definition 3.13. A set E in ${ℝ}^n$ is said to be of (F-)capacity zero, or to have (F-)capacity zero if

$${\text{cap}}_F(E,B)=0$$

whenever E ⊂ B′ ⊂ B. In this case, we write cap_FE = 0.

According to Lemma 3.12 (i), we immediately notice that every single point is of F-capacity zero if α*(F) ≥ 0. Indeed, we are going to show that: to check whether a compact set K is of capacity zero or not, it is enough to test with respect to one ball B (Corollary 3.15). For this purpose, we require the following version of a capacitary estimate, called “comparable lemma”.

Lemma 3.14 (Comparable lemma). If K ⊂ B′ = B_7r/5 and 0 < r ≤ s ≤ 2r, then there exists a universal constant c > 0 such that

$$\frac{1}{c}{\text{cap}}_F\left(K,B_{2r}\right)\le {\text{cap}}_F\left(K,B_{2s}\right)\le c{\text{cap}}_F\left(K,B_{2r}\right).$$

Proof. We may assume x₀ = 0. We claim that for $0<r\le s\le \frac{21}{20}r$, we have

$$\frac{1}{c}{\text{cap}}_F\left(K,B_{2r}\right)\le {\text{cap}}_F\left(K,B_{2s}\right)\le c{\text{cap}}_F\left(K,B_{2r}\right).$$

Indeed, we may iterate this inequality finitely many times to conclude the desired inequality for 0 < r ≤ s ≤ 2r. Moreover, let $y_r=\frac{3}{2}r{e}_1$, $y_s=\frac{3}{2}s{e}_1$ and denote $u_r≔{\widehat{R}}_K^1\left(B_{2r}\right)$, $u_s≔{\widehat{R}}_K^1\left(B_{2s}\right)$.

By the definition of the capacity potential, it is immediate that u_r ≤ u_s in B_2r. In particular, we have

$${\text{cap}}_F\left(K,B_{2r}\right)=u_r\left(y_r\right)\le u_s\left(y_r\right).$$

On the other hand, an application of Harnack inequality for u_s (in a small neighborhood of B_3s/2 \ B_10s/7) yields that there exists a constant c > 0 which is independent of the choice of r and s such that

$$u_s\left(y_r\right)\le c{u}_s\left(y_s\right)=c{\text{cap}}_F\left(K,B_{2s}\right).$$

Here note that $\left|y_r\right|=\frac{3}{2}r\ge \frac{10}{7}s>\frac{7}{5}s$ and $R_K^1\left(B_{2s}\right)$ is F-harmonic in B_2s\B_7s/5 and B_3s/2\B_10s/7 ⊂ B_2s \ B_7s/5. Therefore, it finishes the proof for the first inequality.

Next, for the second inequality, we first assume that α*(F) > 0 and the homogeneous solution is given by $V(x)=|x{|}^{-{\alpha }^{*}}$ (for computational simplicity) and let

$$M≔\underset{\partial B_{2r}}{\text{max\hspace{0.17em}}}{u}_s\in [0,1).$$

Then recalling Theorem 3.6, the comparison principle yields that

$$(1-M)u_r+M\ge u_s\text{ in }{B}_{2r}\backslash K.$$ (3.1)

Now choose z ∈ ∂B_3r/2 so that

$$u_s(z)=\underset{\partial B_{3r/2}}{\text{max\hspace{0.17em}}}{u}_s=:M_1.$$

Then it can be easily checked that the function

$$w(x)≔M_1\cdot \frac{|x{|}^{-{\alpha }^{*}}-{(2s)}^{-{\alpha }^{*}}}{{(3r/2)}^{-{\alpha }^{*}}-{(2s)}^{-{\alpha }^{*}}}$$

is F-harmonic in B_2s \ B_3r/2 and by the comparison principle, w ≥ u_s in B_2s \ B_3r/2. (here again note that $\frac{7}{5}s<\frac{3}{2}r$.) In particular,

$$M_1\cdot \frac{{(2r)}^{-{\alpha }^{*}}-{(2s)}^{-{\alpha }^{*}}}{{(3r/2)}^{-{\alpha }^{*}}-{(2s)}^{-{\alpha }^{*}}}\ge M,$$

$$M_1\cdot \frac{{(3s/2)}^{-{\alpha }^{*}}-{(2s)}^{-{\alpha }^{*}}}{{(3r/2)}^{-{\alpha }^{*}}-{(2s)}^{-{\alpha }^{*}}}\ge u_s\left(\frac{3}{2}s{e}_1\right)={\text{cap}}_F\left(K,B_{2s}\right).$$

Since ${\left(3r/2\right)}^{-{\alpha }^{*}}-{\left(2r\right)}^{-{\alpha }^{*}}\ge {\left(3s/2\right)}^{-{\alpha }^{*}}-{\left(2s\right)}^{-{\alpha }^{*}}$ or equivalently,

$${\left(3r/2\right)}^{-{\alpha }^{*}}-{\left(2s\right)}^{-{\alpha }^{*}}\ge \left[{\left(3s/2\right)}^{-{\alpha }^{*}}-{\left(2s\right)}^{-{\alpha }^{*}}\right]+\left[{\left(2r\right)}^{-{\alpha }^{*}}-{\left(2s\right)}^{-{\alpha }^{*}}\right],$$

we obtain

$$u_s(z)=M_1\ge M+{\text{cap}}_F\left(K,B_{2s}\right).$$ (3.2)

Moreover, by (3.1) and (3.2), we have u_r(z) ≥ (1 − M)u_r(z) ≥ cap_F (K, B_2s) and then Harnack inequality leads to

$${\text{cap}}_F\left(K,B_{2s}\right)\le c{\text{cap}}_F\left(K,B_{2r}\right),$$

for constant c > 0 which is independent of r and s. Finally, for the general homogeneous solution or the case of α*(F) ≤ 0, one can follow the idea of Lemma 3.11. □

Corollary 3.15. Suppose that cap(K, B) = 0 for K ⊂ B′ ⊂ B. Then

1. for any ball B₁ such that $K\subset B_1^{\prime }$ and B₁ ⊂ B′, we have

   $$\text{cap}\left(K,B_1\right)=0;$$
2. for any ball B₂ such that $B_2^{\prime }\supset B$, we have

   $$\text{cap}\left(K,B_2\right)=0;$$

3. K is of F-capacity zero.

Proof.

1. Apply the first inequality of Lemma 3.14 finitely many times.

2. Apply the second inequality of Lemma 3.14 finitely many times.

3. It is an immediate consequence of (i) and (ii).

□

Now we shortly illustrate the potential theoretic meaning of capacity zero sets, at least for convex operators F. In the end, F-capacity zero sets are ‘negligible’ in view of the fully nonlinear operator F; i.e. F-capacity really measures the size of given sets in a suitable way to interpret the corresponding PDE.

Definition 3.16 (Polar sets). A compact set K is called F-polar, or simply polar, if there exist an open ball B_2r with K ⊂ B_7r/5, and F-superharmonic function u in B_2r such that u|_K = ∞.

Lemma 3.17. Suppose that K is a compact set in B_7r/5 and F is convex. Then the followings are equivalent:

1. K is polar.

2. cap_FK = 0.

Proof. (i) ⇒ (ii): Since K is polar, let u be an F-superharmonic function in B_2r such that u|_K = ∞. Recalling the definition of F-superharmonic functions, there exists a point x₀ ∈ B_2r\K such that u(x₀) < ∞. Since u is lower semi-continuous and u cannot attain the value −∞, we may assume inf_B2r u > −∞ by choosing a little smaller ball B_2r′ instead of B_2r. Then by adding a positive constant if necessary, we further assume inf_B2r u ≥ 0, i.e. u is non-negative in B_2r. Note that we still have u is F-superharmonic in B_2r and u|_K = ∞. Therefore, for any ε > 0, we have $\epsilon u\in {\Phi }_K^1\left(B_{2r}\right)$ and so

$${\widehat{R}}_K^1\left(B_{2r}\right)\le \epsilon u.$$

Letting ε → 0 and taking x = x₀, we notice that ${\widehat{R}}_K^1\left(B_{2r}\right)\left(x_0\right)=0$. Finally, the strong minimum principle implies that cap_FK = 0.

(ii) ⇒ (i): Let $y_0=x_0+\frac{3}{2}r{e}_1$. Then by the definition of the capacity and the capacity potential, we have ${\widehat{R}}_K^1\left(B_{2r}\left(x_0\right)\right)\left(y_0\right)=0$. Thus, there exists a sequence of F-superharmonic functions ${\left\{u_j\right\}}_{j=1}^{\infty }$ in B_2r such that

$$u_j\ge 0\text{ in }{B}_{2r},u_j\ge 1\text{ on }K\text{ and }{u}_j\left(y_0\right)<1/2^j\text{.}$$

Define $v_k≔{\sum }_{j=1}^k{u}_j$ which is lower semi-continuous and is finite in a dense subset of Ω. Furthermore, since F is convex, we have F(D²v_k) ≤ 0, and so v_k is F-superharmonic. Since {v_k}_k is an increasing sequence of F-superharmonic functions, Lemma 2.8 (iii) gives that the limit function v = v_k is either F-superharmonic or v ≡ ∞. The second possibility is excluded because 0 ≤ v(y₀) ≤ 1. Therefore, v is F-superharmonic in B_2r and v|_K = ∞, which implies that K is polar.

□

Definition 3.18 (Removable sets). A compact set K(⊂ B_7r/5) is called F-removable, or simply removable, if for each function u that is F-superharmonic on B_2r \ K and is bounded below in a neighborhood of K, there exists an extension U of u which is F-superharmonic in B_2r and U = u in B_2r \ K.

Lemma 3.19. Suppose that K is a compact set of capacity zero and F is convex. Then K is removable.

Proof. Let u be an F-superharmonic function in B_2r \ K and is bounded below in a neighborhood of K. Since K is of capacity zero, we have ${\widehat{R}}_K^1\left(B_{2r}\right)\left(y_0\right)=0$ and so ${\widehat{R}}_K^1\left(B_{2r}\right)=0$ by the strong minimum principle. In particular, $R_K^1\left(B_{2r}\right)\equiv 0$ in B_2r \ K. Now, for any z₀ ∈ B_2r \ K, following the proof of [(ii) ⇒ (i)] part in Lemma 3.17, there exists a non-negative F-superharmonic function $v_{z_0}$ in B_2r such that $v_{z_0}{|}_K=\infty$ and $v_{z_0}\left(z_0\right)<\infty$.

Now we consider a canonical lower semi-continuous extension U of u across K, which is defined by

$$U(x)=\{\begin{array}{ll}{\text{lim inf }}_{y\to x,y\notin K}u(y)\hfill & \text{if }x\notin \text{int }K,\hfill \\ \infty \hfill & \text{if }x\in \text{int }K.\hfill \end{array}$$

Then U is the lower semi-continuous regularization of the function v, where

$$v=\{\begin{array}{ll}u\hfill & \text{in }{B}_{2r}\backslash K,\hfill \\ \infty \hfill & \text{on }K.\hfill \end{array}$$

See [11] for details. Moreover, by Lemma 2.2 and Lemma 2.10, we notice that U = u in B_2r \ K and so U is F-superharmonic in B_2r \ K.

Then we claim that $U+\epsilon v_{z_0}$ is F-superharmonic in B_2r, for any ε > 0 and z₀ ∈ B_2r\K. Indeed, the convexity of F immediately guarantees that $U+\epsilon v_{z_0}$ is F-superharmonic in B_2r \ K. On the other hand, since $U+\epsilon v_{z_0}{|}_K=\infty$, we cannot choose any test functions for $U+\epsilon v_{z_0}$ at points in K. In other words, for any φ ∈ C²(Ω), $U+\epsilon v_{z_0}-\phi$ cannot have a local minimum at x₀ ∈ K. Thus, recalling the equivalence of F-supersolution and F-superharmonic function (Theorem 2.10), we conclude that $U+\epsilon v_{z_0}$ is F-superharmonic in B_2r.

Now let $ℱ={\left\{U+\epsilon v_{z_0}\right\}}_{\epsilon >0,z_0\in B_{2r}\backslash K}$ be a family of F-superharmonic functions in B_2r. Since u is bounded below in a neighborhood of K and $v_{z_0}$ is non-negative, any element in ℱ is locally uniformly bounded below. Thus, applying Lemma 3.1, we have

$$s(x)=\underset{r\to 0}{\text{lim}}\underset{B_r(x)}{\text{inf\hspace{0.17em}}}(\text{inf }ℱ)$$

is F-superharmonic in B_2r. On the other hand, it is easy to check that

$$\text{inf }ℱ=\{\begin{array}{ll}u\hfill & \text{in }{B}_{2r}\backslash K,\hfill \\ \infty \hfill & \text{on }K.\hfill \end{array}$$

Therefore, we conclude that s = U and U is a desired extension of u. □

Remark 3.20. Considering the dual operator $\tilde{F}$, one can obtain analogous definitions and corresponding results when the operator is concave.

For similar results concerning polar sets and removable sets, see [12] for p-Laplacian operators, [24] for Pucci extremal operators, and [23] for k-Hessian operators. See also [2, 10, 11] for the analysis of polar sets and removable sets in view of Riesz capacity or Hausdorff measure.

3.4. Another characterization of a regular point.

The definitions of a reduced function and a balayage depend on the choice of an operator F. In this subsection, we need to distinguish an operator and its dual operator, so we will specify the dependence by denoting ${\widehat{R}}_K^{1,F}(\Omega )$ or ${\widehat{R}}_K^{1,\tilde{F}}(\Omega )$. We now provide a key lemma for our first main theorem, the sufficiency of the Wiener criterion:

Lemma 3.21. A boundary point x₀ ∈ ∂Ω is regular if

$${\widehat{R}}_{\overline{B}\backslash \Omega }^{1,\tilde{F}}(2B)\left(x_0\right)=1={\widehat{R}}_{\overline{B}\backslash \Omega }^{1,F}(2B)\left(x_0\right)$$

whenever B is a ball centered at x₀.

Proof. For f ∈ C(∂Ω), consider the upper Perron solution ${\overline{H}}_f={\overline{H}}_f(\Omega )$. We may assume f(x₀) = 0 and max_∂Ω |f| ≤ 1. For ε > 0, we can choose a ball B with center x₀ such that ∂(2B) ∩ Ω ≠ ∅ and |f| < ε in 2B ∩ ∂Ω. Then we define

$$u=\{\begin{array}{ll}1+\epsilon -{\widehat{R}}_{\overline{B}\backslash \Omega }^{1,\tilde{F}}(2B)\hfill & \text{in }\Omega \cap 2B,\hfill \\ 1+\epsilon \hfill & \text{in }\Omega \backslash 2B.\hfill \end{array}$$

Since ${\widehat{R}}_{\overline{B}\backslash \Omega }^{1,\tilde{F}}(2B)$ is a $\tilde{F}$-solution in Ω ∩ 2B, $1+\epsilon -{\widehat{R}}_{\overline{B}\backslash \Omega }^{1,\tilde{F}}(2B)$ is F-harmonic in Ω ∩ 2B. On the other hand, by Theorem 3.6, ${\widehat{R}}_{\overline{B}\backslash \Omega }^{1,\tilde{F}}(2B)$ can be considered as the upper Perron solution for the operator $\tilde{F}$. Then since a ball is regular, we have

$$\underset{y\to x}{\text{lim}}{\widehat{R}}_{\overline{B}\backslash \Omega }^{1,\tilde{F}}(2B)(y)=0\text{ for all }x\in \partial (2B)\text{.}$$

Thus, u is continuous in Ω and by the pasting lemma, u is F-superharmonic in Ω. Moreover, it can be easily checked that

$$\underset{y\to x}{\text{lim inf }}u(y)\ge f(x)\text{ for any }x\in \partial \Omega .$$

Therefore, u ∈ 𝒰_f and so ${\overline{H}}_f\le u$. In particular,

$$\underset{\Omega \ni y\to x_0}{\text{lim sup }}{\overline{H}}_f(y)\le \underset{\Omega \ni y\to x_0}{\text{lim sup }}u(y)=1+\epsilon -\underset{\Omega \ni y\to x_0}{\text{lim inf }}{\widehat{R}}_{\overline{B}\backslash \Omega }^{1,\tilde{F}}(2B)(y)\le 1+\epsilon -{\widehat{R}}_{\overline{B}\backslash \Omega }^{1,\tilde{F}}(2B)\left(x_0\right)=\epsilon .$$

For the converse inequality, we define

$$v=\{\begin{array}{ll}-1-\epsilon +{\widehat{R}}_{\overline{B}\backslash \Omega }^{1,F}(2B)\hfill & \text{in }\Omega \cap 2B,\hfill \\ -1-\epsilon \hfill & \text{in }\Omega \backslash 2B.\hfill \end{array}$$

Then by a similar argument, v ∈ ℒ_f and so,

$$\underset{\Omega \ni y\to x_0}{\text{lim inf }}{\underset{\_}{H}}_f(y)\ge -\epsilon .$$

Consequently, since ε > 0 is arbitrary, we conclude that

$$\underset{\Omega \ni y\to x_0}{\text{lim}}{\overline{H}}_f(y)=\underset{\Omega \ni y\to x_0}{\text{lim}}{\underset{\_}{H}}_f(y)=0=f\left(x_0\right),$$

i.e. x₀ is regular. □

Next, we exhibit a converse direction of the above lemma: i.e. a characterization of an irregular boundary point. We expect that this lemma may be employed to prove the necessity of the Wiener criterion for the general case.

Lemma 3.22 (Characterization of an irregular boundary point). If there exists a constant ρ > 0 such that the capacity potential u = u_ρ of $\overline{B_{\rho }\left(x_0\right)}\backslash \Omega$ with respect to B_2ρ(x₀) satisfies the inequality

$$u\left(x_0\right)={\widehat{R}}_{\overline{B_{\rho }\left(x_0\right)}\backslash \Omega }^1\left(B_{2\rho }\left(x_0\right)\right)<1,$$

then the boundary point x₀ ∈ ∂Ω is irregular.

Proof. Since the capacity potential u is the lower semi-continuous regularization, we have

$$u\left(x_0\right)=\underset{\Omega \ni x\to x_0}{\text{lim inf }}u(x)<1.$$ (3.3)

Moreover, by definition, we have u_ρ′ ≤ u_ρ when 0 < ρ′ < ρ. Thus, we can choose a sufficiently small ρ > 0 such that (3.3) holds and Ω ∩ ∂B_2ρ(x₀) ≠ ∅.

Now we define a smooth boundary data f on ∂(Ω ∩ B_2ρ(x₀)) such that f(x) = 3/2 if x ∈ ∂Ω ∩ B_ρ/2(x₀), 0 ≤ f(x) ≤ 3/2 if x ∈ ∂Ω ∩ (B_ρ(x₀) \ B_ρ/2(x₀)) and f(x) = 0 on the remaining part of ∂(Ω ∩ B_2ρ(x₀)). Then we consider the lower Perron solution ${\underset{\_}{H}}_f\left(\Omega \cap B_{2\rho }\left(x_0\right)\right)$. We claim that the following inequality holds:

$${\underset{\_}{H}}_f(x)\le \frac{1}{2}+u(x),x\in \Omega \cap B_{2\rho }\left(x_0\right).$$ (3.4)

Recalling the comparison principle, it is enough to check the above inequality on the boundary of the domain Ω∩B_2ρ(x₀). For this purpose, let v ∈ ℒ_f(Ω∩B_2ρ(x₀)) and $w\in {𝒰}_g\left(B_{2\rho }(x_0)\backslash \left(\overline{B_{\rho }\left(x_0\right)}\backslash \Omega \right)\right)$ where g is given by (recall Theorem 3.6)

$$g=\{\begin{array}{ll}1\hfill & \text{on }\partial \left(\overline{B_{\rho }\left(x_0\right)}\backslash \Omega \right),\hfill \\ 0\hfill & \text{in }\partial B_{2\rho }\left(x_0\right).\hfill \end{array}$$

1. (on ∂Ω ∩ B_2ρ(x₀)) First, for x ∈ ∂Ω ∩ B_ρ(x₀), we have

   $$\underset{y\to x}{\text{lim sup }}v(y)\le f(x)\le \frac{3}{2}=\frac{1}{2}+g(x)\le \frac{1}{2}+\underset{y\to x}{\text{lim inf }}w(y).$$
   Next, for x ∈ ∂Ω ∩ (B_2ρ(x₀) \ B_ρ(x₀)), we have

   $$\underset{y\to x}{\text{lim sup }}v(y)\le f(x)=0\le \frac{1}{2}+g(x)\le \frac{1}{2}+\underset{y\to x}{\text{lim inf }}w(y).$$
2. (on Ω ∩ ∂B_2ρ(x₀)) Similarly, we obtain

   $$\underset{y\to x}{\text{lim sup }}v(y)\le f(x)=0\le \frac{1}{2}+g(x)\le \frac{1}{2}+\underset{y\to x}{\text{lim inf }}w(y).$$

Now since v and w are F-subharmonic and F-superharmonic, respectively, we derive that

$$v\le \frac{1}{2}+w,\text{ in }\Omega \cap B_{2\rho }\left(x_0\right).$$

Taking the supremum on v and the infimum on w, we conclude (3.4) which implies that

$$\underset{\Omega \cap B_{2\rho }\left(x_0\right)\ni x\to x_0}{\text{lim inf }}{\underset{\_}{H}}_f(x)\le \frac{1}{2}+\underset{\Omega \cap B_{2\rho }\left(x_0\right)\ni x\to x_0}{\text{lim inf }}u(x)<\frac{3}{2}=f\left(x_0\right).$$

Therefore, x₀ is irregular with respect to Ω ∩ B_2ρ(x₀). Recalling Lemma 2.24, we deduce that x₀ is irregular with respect to Ω. □

4. A sufficient condition for the regularity of a boundary point

In this section, we prove the sufficiency of the Wiener criterion and its sequential corollaries, via the potential estimates. More precisely, we first develop quantitative estimates for the capacity potential ${\widehat{R}}_K^1(B)$ by employing capacitary estimates obtained in Section 3. Then we adopt the characterization of a regular boundary point in terms of the capacity potential to deduce the desired conclusion.

Definition 4.1. We say that a set E is F-thick at z if the Wiener integral diverges, i.e.

$${\int }_0^1{\text{cap}}_F\left(E\cap \overline{B_t(z)},B_{2t}(z)\right)\frac{\text{d}t}{t}=\infty .$$ (4.1)

For simplicity, we write

$${\phi }_F(z,E,t)={\text{cap}}_F\left(E\cap \overline{B_t(z)},B_{2t}(z)\right),$$

for the capacity density function in (4.1).

Remark 4.2. Recalling Lemma 3.11, there exists a constant c > 0 which is independent of t > 0 such that

$$1/c\le {\text{cap}}_F\left(\overline{B_t},B_{2t}\right)\le c.$$

Thus, one may write an equivalent form of (4.1):

$${\int }_0^1\frac{{\text{cap}}_F\left(E\cap \overline{B_t(z)},B_{2t}(z)\right)}{{\text{cap}}_F\left(\overline{B_t(z)},B_{2t}(z)\right)}\frac{\text{d}t}{t}=\infty ,$$

which is a similar form to the Wiener integral appearing in [20, 40].

Now we can state an equivalent form of our main theorem, Theorem 1.2:

If Ω^c is both F-thick and $\tilde{F}$-thick at a boundary point x₀ ∈ ∂Ω, then x₀ is regular. To prove this statement, we need several auxiliary lemmas regarding the capacity potential.

Lemma 4.3. Fix a ball B. Suppose that K ⊂ B′ is compact and $v={\widehat{R}}_K^1(B)$. If 0 < γ < 1 and K_γ ≔ {x ∈ B : v(x) ≥ γ} ⊂ B′, then

$$\text{cap}\left(K_{\gamma },B\right)=\frac{1}{\gamma }\text{cap}(K,B)$$

Proof. We write $v_{\gamma }≔{\widehat{R}}_{K_{\gamma }}^1(B)$. Then by Lemma 3.4 and the definition of a reduced function,

$$v_{\gamma }=R_{K_{\gamma }}^1\left(B\right)=\text{inf }{\Phi }_{K_{\gamma }}^1=\text{inf}\left\{w:w\text{ is }F\text{-superharmonic in }B,w\ge {\psi }_{\gamma }\text{ in }B\right\}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}in }B\backslash K_{\gamma },$$

where

$${\psi }_{\gamma }=\{\begin{array}{ll}1\hfill & \text{in }{K}_{\gamma },\hfill \\ 0\hfill & \text{in }B\backslash K_{\gamma }.\hfill \end{array}$$

1. Clearly, $v={\widehat{R}}_K^1(B)$ is F-superharmonic in B and so is v/γ due to (F2). Since v ≥ γ in K_γ, we have v/γ ≥ 1 in K_γ. Thus, $v/\gamma \in {\Phi }_{K_{\gamma }}^1$ and so

   $$\frac{v}{\gamma }\ge v_{\gamma }\text{ in }B\backslash K_{\gamma }.$$
2. Recalling Theorem 3.6, $v_{\gamma }={\overline{H}}_{f_{\gamma }}\left(B\backslash K_{\gamma }\right)$ in B\K_γ where

   $$f_{\gamma }=\{\begin{array}{ll}1\hfill & \text{on }\partial K_{\gamma },\hfill \\ 0\hfill & \text{on }\partial B.\hfill \end{array}$$
   Then for $u\in {𝒰}_{f_{\gamma }}\left(B\backslash K_{\gamma }\right)$, we have

   $$\underset{B\backslash K_{\gamma }\ni y\to x}{\text{lim inf }}u(y)\ge f_{\gamma }(x)=1=\frac{v(x)}{\gamma },$$

   for any x ∈ ∂K_γ. Since u is F-superharmonic and v/γ is F-harmonic in B \ K_γ, the comparison principle leads to u ≥ v/γ in B \ K_γ and so

   $$v_{\gamma }\ge \frac{v}{\gamma }\text{ in }B\backslash K_{\gamma }.$$

Consequently, we conclude that

$$\text{cap}\left(K_{\gamma },B\right)=v_{\gamma }\left(y_0\right)=\frac{1}{\gamma }v\left(y_0\right)=\frac{1}{\gamma }\text{cap}(K,B).$$

□

Lemma 4.4. Fix a ball B = B_2r(x₀). Let K ⊂ B_r = B_r(x₀) be a compact set and $v={\widehat{R}}_K^1(B)$. Then there exists a constant c > 0 which is independent of K and r such that

$$v(x)\ge c\text{ cap}(K,B),$$

for any x ∈ B_r.

Proof. Denote

$$M≔\underset{\partial B_{6r/5}}{\text{sup\hspace{0.17em}}}v,m≔\underset{\partial B_{6r/5}}{\text{inf\hspace{0.17em}}}v.$$

Since v is a non-negative F-solution in B\K, Harnack inequality yields that there exists a constant c₁ > 0 independent of r > 0 such that

$$c_{1}M\le m.$$ (4.2)

Morevoer, the strong maximum principle in B \ B_6r/5 implies that

$$K_M≔\left\{v\in B:v(x)\ge M\right\}\subset \overline{B_{6r/5}},$$

and so

$$\text{cap}\left(K_M,B\right)\le \text{cap}\left(\overline{B_{6r/5}},B\right)~1.$$ (4.3)

Here we applied Lemma 3.11 and the comparable constant does not depend on K and r.

Now since K_M ⊂ B′, we can apply Lemma 4.3:

$$\text{cap}\left(K_M,B\right)=\frac{1}{M}\text{cap}(K,B).$$ (4.4)

Finally, combining (4.2), (4.3) and (4.4), we conclude that

$$m\ge c_{1}M=c_1\cdot \frac{\text{cap}(K,B)}{\text{cap}\left(K_M,B\right)}\ge c_2\text{cap}(K,B),$$

and the minimum principle leads to the desired result. □

We may rewrite the previous lemma as

$${\widehat{R}}_K^1\left(B_{2r}\right)(x)\ge c{\phi }_F\left(x_0,K,r\right),\text{ for any }x\in B_r.$$ (4.5)

Lemma 4.5. Let x₀ ∈ ∂Ω, ρ > 0 and

$$w=1-\widehat{R}\frac{1}{B_{\rho }\left(x_0\right)\backslash \Omega }\left(B_{2\rho }\left(x_0\right)\right).$$

Then for all 0 < r ≤ ρ, there exists a constant c > 0 such that

$$w(x)\le \text{exp}\left(-c{\int }_r^{\rho }{\phi }_F\left(x_0,{\Omega }^c,t\right)\frac{\text{d}t}{t}\right),$$

for any x ∈ B_r(x₀).

Proof. Denote $B_i=B_{2^{1-i}\rho }\left(x_0\right)$. Fix 0 < r ≤ ρ and let k be the integer with 2^−kρ < r ≤ 2^1−kρ.

Then write for i = 0, 1, 2, …

$$v_i≔{\widehat{R}}_{\frac{1}{B_{i+1}}\backslash \Omega }\left(B_i\right)$$

and

$$a_i≔{\phi }_F\left(x_0,{\Omega }^c,2^{-i}\rho \right).$$

Since e^t ≥ 1 + t, estimate (4.5) yields that

$$v_i\ge c{a}_i\ge 1-\text{exp}\left(-c{a}_i\right)\text{ in }{B}_{i+1}.$$

Thus, denoting m₀ ≔ inf_B1 v₀, we have

$$1-m_0\le \text{exp}\left(-c{a}_0\right).$$

Next, let $D_1≔B_1\backslash \left(\overline{B_2}\cap {\Omega }^c\right)$ and

$${\psi }_1≔\{\begin{array}{ll}1\hfill & \text{in }\overline{B_2}\cap {\Omega }^c,\hfill \\ m_0\hfill & \text{in }{D}_1.\hfill \end{array}$$

Then we write $u_1≔{\widehat{R}}^{{\psi }_1}\left(B_1\right)$ be the balayage with respect to the ψ₁ in B₁. It immediately follows from the definition of balayage that

$$\frac{u_1-m_0}{1-m_0}={\widehat{R}}_{{\overline{B}}_2\cap {\Omega }^c}^1\left(B_1\right)=v_1.$$

Again, denoting $m_1≔{\text{inf\hspace{0.17em}}}_{B_2}{u}_1$, we obtain

$$1-m_1\le \left(1-m_0\right)\text{exp}\left(-c{a}_1\right)\le \text{exp}\left(-c\left(a_1+a_0\right)\right).$$

Now iterate this step: let $D_i≔B_i\backslash \left(\overline{B_{i+1}}\cap {\Omega }^c\right)$ and

$${\psi }_i=\{\begin{array}{ll}1\hfill & \text{in }\overline{B_{i+1}}\cap {\Omega }^c,\hfill \\ m_{i-1}\hfill & \text{in }{D}_i.\hfill \end{array}$$

Denoting $u_i≔{\widehat{R}}^{{\psi }_i}\left(B_i\right)$ and $m_i≔{\text{inf\hspace{0.17em}}}_{B_{i+1}}{u}_i$, we have

$$\frac{u_i-m_{i-1}}{1-m_{i-1}}=v_i$$

and so

$$1-m_i\le \left(1-m_{i-1}\right)\text{exp}\left(-c{a}_i\right)\le \text{exp}\left(-c\sum _{j=0}^i{a}_j\right).$$

Furthermore, we claim that u_i ≥ u_i+1 in B_i+1. Indeed, by Theorem 3.6, $u_i={\overline{H}}_{f_i}\left(D_i\right)$ in D_i where f_i ∈ C(∂D_i) is given by

$$f_i=\{\begin{array}{ll}1\hfill & \text{in }\partial \left(\overline{B_{i+1}}\cap {\Omega }^c\right),\hfill \\ m_{i-1}\hfill & \text{in }\partial B_i.\hfill \end{array}$$

Thus, for $u\in {𝒰}_{f_i}\left(D_i\right)$, we have

$$\underset{D_{i+1}\ni y\to x}{\text{lim inf }}u(y)\ge 1\ge \underset{D_{i+1}\ni y\to x}{\text{lim sup }}{u}_{i+1}(y)\text{ for any }x\in \partial \left(\overline{B_{i+2}}\cap {\Omega }^c\right),$$

$$\underset{D_{i+1}\ni y\to x}{\text{lim inf }}u(y)\ge m_i=\underset{D_{i+1}\ni y\to x}{\text{lim sup }}{u}_{i+1}(y)\text{ for any }x\in \partial B_{i+1}\text{.}$$

Therefore, by the comparison principle, u ≥ u_i+1 in D_i+1 and so $u_i={\overline{H}}_{f_i}\left(D_i\right)\ge u_{i+1}$ in B_i+1.

Repeating the argument above, we conclude that v₀ ≥ u₁ ≥ ⋯ ≥ u_k in B_k, which implies that

$$w=1-v_0\le 1-u_k\le 1-m_k\le \text{exp}\left(-c\sum _{j=0}^k{a}_j\right)\text{in }{B}_{k+1}.$$

Finally, the result follows from

$${\int }_r^{\rho }{\phi }_F\left(x_0,{\Omega }^c,t\right)\frac{\text{d}t}{t}\le c\sum _{i=1}^k{a}_i,$$

which can be easily checked from the dyadic decomposition. Indeed, we can deduce from Lemma 3.11 and Lemma 3.14 that if t ≤ s ≤ 2t, then

$${\text{cap}}_F\left(\overline{B_t}\backslash K,B_{2t}\right)~{\text{cap}}_F\left(\overline{B_t}\backslash K,B_{2s}\right),$$

where the comparable constant only depends on n, λ, Λ and these results also hold for ${\text{cap}}_{\tilde{F}}(\cdot )$. □

Now we are ready to prove the sufficiency of the Wiener criterion, Theorem 1.2.

Proof of Theorem 1.2. Let x₀ ∈ ∂Ω, ρ > 0 and define

$$w_{F,\rho }≔1-{\widehat{R}}_{\overline{B_{\rho }\left(x_0\right)}\backslash \Omega }^{1,F}\left(B_{2\rho }\left(x_0\right)\right)\text{ and }{w}_{\tilde{F},\rho }≔1-{\widehat{R}}_{\overline{B_{\rho }\left(x_0\right)}\backslash \Omega }^{1,\tilde{F}}\left(B_{2\rho }\left(x_0\right)\right).$$

Then applying Lemma 4.5 for both functions, we have that for all 0 < r ≤ ρ, there exist a constant c₁, c₂ > 0 such that

$$w_{F,\rho }(x)\le \text{exp}\left(-c_1{\int }_r^{\rho }{\phi }_F\left(x_0,{\Omega }^c,t\right)\frac{\text{d}t}{t}\right),$$

$$w_{\tilde{F},\rho }(x)\le \text{exp}\left(-c_2{\int }_r^{\rho }{\phi }_{\tilde{F}}\left(x_0,{\Omega }^c,t\right)\frac{\text{d}t}{t}\right),$$

for any x ∈ B_r(x₀). Letting r → 0⁺, we conclude that

$${\widehat{R}}_{\overline{B_{\rho }\left(x_0\right)}\backslash \Omega }^{1,F}\left(B_{2\rho }\left(x_0\right)\right)\left(x_0\right)=1={\widehat{R}}_{\overline{B_{\rho }\left(x_0\right)}\backslash \Omega }^{1,\tilde{F}}\left(B_{2\rho }\left(x_0\right)\right)\left(x_0\right).$$

Since ρ > 0 can be arbitrarily chosen, an application of Lemma 3.21 yields that x₀ ∈ ∂Ω is a regular boundary point. (Note that a boundary point x₀ is F-regular if and only if it is $\tilde{F}$-regular; Corollary 2.25.) □

On the other hand, if additional information is imposed on the boundary data f, i.e. the boundary data f has its maximum (or minimum) at x₀ ∈ ∂Ω, then we can deduce the continuity of the Perron solution at x₀ under a relaxed condition:

Corollary 4.6. Suppose that f ∈ C(∂Ω) attains its maximum [resp. minimum] at x₀ ∈ ∂Ω. If Ω^c is F-thick [resp. $\tilde{F}$-thick] at x₀ ∈ ∂ Ω, then

$$\underset{\Omega \ni y\to x_0}{\text{lim}}{\overline{H}}_f(y)=f\left(x_0\right)=\underset{\Omega \ni y\to x_0}{\text{lim}}{\underset{\_}{H}}_f(y).$$

Proof. Similarly as in the proof of the previous theorem, this corollary is the consequence of Lemma 3.21 and Lemma 4.5. □

Furthermore, if the given boundary data f ∈ C(∂Ω) is resolutive, then we are able to obtain a quantitative estimate for the modulus of continuity.

Lemma 4.7 (The modulus of continuity). Suppose that Ω is an open and bounded subset of ${ℝ}^n$. Let f ∈ C(∂Ω).

If x₀ ∈ ∂Ω with f(x₀) = 0, then for 0 < r ≤ ρ, we have

$$\underset{{\Omega }_r}{\text{sup\hspace{0.17em}}}{\underset{\_}{H}}_f^F\le \underset{\partial {\Omega }_{2\rho }}{\text{max\hspace{0.17em}}}f+\underset{\partial \Omega }{\text{max\hspace{0.17em}}}f\cdot \text{exp}\left(-c{\int }_r^{\rho }{\phi }_{\tilde{F}}\left(x_0,{\Omega }^c,t\right)\frac{\text{d}t}{t}\right)$$

and

$$\underset{{\Omega }_r}{\text{inf\hspace{0.17em}}}{\overline{H}}_f^F\ge \underset{\partial {\Omega }_{2\rho }}{\text{min\hspace{0.17em}}}f+\underset{\partial \Omega }{\text{min\hspace{0.17em}}}f\cdot \text{exp}\left(-c{\int }_r^{\rho }{\phi }_F\left(x_0,{\Omega }^c,t\right)\frac{\text{d}t}{t}\right)$$

where Ω_r ≔ Ω ∩ B_r(x₀) and ∂Ω_2ρ ≔ ∂Ω ∩ B_2ρ(x₀).

Furthermore, if f is resolutive, then we have the quantitative estimate:

$$\underset{\partial {\Omega }_{2\rho }}{\text{min\hspace{0.17em}}}f+\underset{\partial \Omega }{\text{min\hspace{0.17em}}}f\cdot \text{exp}\left(-c{\int }_r^{\rho }{\phi }_F\left(x_0,{\Omega }^c,t\right)\frac{\text{d}t}{t}\right)\le \underset{{\Omega }_r}{\text{inf\hspace{0.17em}}}{H}_f^F\le \underset{{\Omega }_r}{\text{sup\hspace{0.17em}}}{H}_f^F\le \underset{\partial {\Omega }_{2\rho }}{\text{max\hspace{0.17em}}}f+\underset{\partial \Omega }{\text{max\hspace{0.17em}}}f\cdot \text{exp}\left(-c{\int }_r^{\rho }{\phi }_{\tilde{F}}\left(x_0,{\Omega }^c,t\right)\frac{\text{d}t}{t}\right),$$

where $H_f^F≔{\overline{H}}_f^F={\underset{\_}{H}}_f^F$.

Proof. Let $v={\widehat{R}}_{\overline{B_{\rho }\left(x_0\right)}\backslash \Omega }^{1,\tilde{F}}\left(B_{2\rho }\left(x_0\right)\right)$ be the capacity potential of $\overline{B_{\rho }}\backslash \Omega$ with respect to B_2ρ.

Then let w ≔ 1 − v and write

$$s≔w\cdot \underset{\partial \Omega }{\text{max\hspace{0.17em}}}f+\underset{\partial {\Omega }_{2\rho }}{\text{max\hspace{0.17em}}}f.$$

Note that since we assumed f(x₀) = 0, we have max_∂Ω f ≥ 0 and ${\text{max\hspace{0.17em}}}_{\partial {\Omega }_{2\rho }}f\ge 0$. For $u\in {ℒ}_f^F$, u is F-subharmonic and s is F-harmonic in Ω_2ρ. Moreover,

$$\underset{y\to x}{\text{lim inf }}s(y)\ge \underset{\partial {\Omega }_{2\rho }}{\text{max\hspace{0.17em}}}f\ge \underset{y\to x}{\text{lim sup }}u(y)\text{ for any }x\in \partial \Omega \cap B_{2\rho }$$

and

$$\underset{y\to x}{\text{lim inf }}s(y)\ge \underset{\partial \Omega }{\text{max\hspace{0.17em}}}f\ge \underset{y\to x}{\text{lim sup }}u(y)\text{ for any }x\in \Omega \cap \partial B_{2\rho }\text{.}$$

Thus, the comparison principle yields that s ≥ u in Ω_2ρ and so $s\ge {\underset{\_}{H}}_f^F$ in Ω_2ρ.

On the other hand, let

$$\tilde{s}≔\left(1-{\widehat{R}}_{\overline{B_{\rho }\left(x_0\right)}\backslash \Omega }^{1,F}\left(B_{2\rho }\left(x_0\right)\right)\right)\cdot \underset{\partial \Omega }{\text{max\hspace{0.17em}}}(-f)+\underset{\partial {\Omega }_{2\rho }}{\text{max\hspace{0.17em}}}(-f).$$

By the same argument, we derive $\tilde{s}\ge {\underset{\_}{H}}_{-f}^{\tilde{F}}=-{\overline{H}}_f^F$ in ∂Ω_2ρ.

An application of Lemma 4.5 for w (and $\tilde{w}$) finishes the proof. □

Now we present a new geometric condition for a regular boundary point, namely the exterior corkscrew condition; see also [16, 28].

Definition 4.8. We say that Ω satisfies the exterior corkscrew condition at x₀ ∈ ∂Ω if there exists 0 < δ < 1/4 and R > 0 such that for any 0 < r < R, there exists y ∈ B_r(x₀) such that $\overline{B_{\delta r}(y)}\subset {\Omega }^c\cap B_r\left(x_0\right)$.

Note that if Ω satisfies an exterior cone condition at x₀ ∈ ∂Ω, then Ω satisfies an exterior corkscrew condition at x₀. Thus, the following corollary obtained from the (potential theoretic) Wiener criterion is a generalized result of Theorem 2.26.

Corollary 4.9 (Exterior corkscrew condition). Suppose that Ω satisfies an exterior corkscrew condition at x₀ ∈ ∂Ω. Then x₀ is a regular boundary point. Moreover, if f is Hölder continuous at x₀ and is resolutive, then H_f is Hölder continuous at x₀.

Proof. A small modification of Lemma 3.11 and its proof, we have

$$\text{cap}\left(\overline{B_{\delta r}(y)},B_{2r}\left(x_0\right)\right)~1,\text{ for }\delta \in (0,1/4)\text{ and }\overline{B_{\delta r}(y)}\subset B_{2r}\left(x_0\right)\text{,}$$

where the comparable constant depends only on n, λ, Λ and δ. Thus, if x₀ satisfies an exterior corkscrew condition, then we have

$${\int }_0^1{\text{cap}}_F\left(\overline{B_t\left(x_0\right)}\backslash \Omega ,B_{2t}\left(x_0\right)\right)\frac{\text{d}t}{t}\ge {\int }_0^1{\text{cap}}_F\left(\overline{B_{\delta t}(y)},B_{2t}\left(x_0\right)\right)\frac{\text{d}t}{t}\ge \frac{1}{c}{\int }_0^1\frac{\text{d}t}{t}=\infty ,$$

$${\int }_0^1{\text{cap}}_{\tilde{F}}\left(\overline{B_t\left(x_0\right)}\backslash \Omega ,B_{2t}\left(x_0\right)\right)\frac{\text{d}t}{t}\ge {\int }_0^1{\text{cap}}_{\tilde{F}}\left(\overline{B_{\delta t}(y)},B_{2t}\left(x_0\right)\right)\frac{\text{d}t}{t}\ge \frac{1}{c}{\int }_0^1\frac{\text{d}t}{t}=\infty ,$$

and so x₀ is a regular boundary point by the Wiener criterion.

Next, for the second statement, we may assume f(x₀) = 0 by adding a constant for f, if necessary. Since f is resolutive, we can apply the quantitative estimate obtained in Lemma 4.7:

$$\underset{\partial {\Omega }_{2\rho }}{\text{min\hspace{0.17em}}}f+\underset{\partial \Omega }{\text{min\hspace{0.17em}}}f\cdot \text{exp}\left(-c{\int }_r^{\rho }{\phi }_F\left(x_0,{\Omega }^c,t\right)\frac{\text{d}t}{t}\right)\le \underset{{\Omega }_r}{\text{inf\hspace{0.17em}}}{H}_f^F\le \underset{{\Omega }_r}{\text{sup\hspace{0.17em}}}{H}_f^F\le \underset{\partial {\Omega }_{2\rho }}{\text{max\hspace{0.17em}}}f+\underset{\partial \Omega }{\text{max\hspace{0.17em}}}f\cdot \text{exp}\left(-c{\int }_r^{\rho }{\phi }_{\tilde{F}}\left(x_0,{\Omega }^c,t\right)\frac{\text{d}t}{t}\right),$$

Here

1. f is Hölder continuous at x₀: there exists a constant C > 0 such that |f(x)| = |f(x) − f(x₀)| ≤ C|x − x₀|^γ ≤ Cρ^γ for x ∈ ∂Ω_2ρ.

2. Ω satisfies an exterior corkscrew condition at x₀:

   $$\text{exp}\left(-c{\int }_r^{\rho }{\phi }_{\tilde{F}}\left(x_0,{\Omega }^c,t\right)\frac{\text{d}t}{t}\right)\le \text{exp}\left(-c_1{\int }_r^{\rho }\frac{\text{d}t}{t}\right)={\left(\frac{r}{\rho }\right)}^{c_1}.$$

Thus, choosing ρ = r^1/2, we conclude that the Perron solution H_f is Hölder continuous at x₀. □

Remark 4.10 (Example). In this example, we suppose n = 2, $F={𝒫}_{\lambda ,\Lambda }^{+}$ with ellipticity constants 0 < λ < Λ. Then it immediately follows that

$$\tilde{F}={𝒫}_{\lambda ,\Lambda }^{-},{\alpha }^{*}(F)=(n-1)\frac{\lambda }{\Lambda }-1<0,{\alpha }^{*}(\tilde{F})=(n-1)\frac{\Lambda }{\lambda }-1>0.$$

We consider a domain $\Omega =B_1(0)\backslash \left\{0\right\}\subset {ℝ}^2$ and its boundary point 0 ∈ ∂Ω.

1. Since α*(F) < 0, we know that a single point has non-zero capacity. More precisely, recalling the homogeneous solution for F is given by

   $$V(x)=-|x{|}^{1-\frac{\lambda }{\Lambda }},$$

   there exists a constant c = c(λ, Λ) > 0 such that

   $${\text{cap}}_F\left(\left\{0\right\},B_{2t}(0)\right)=c.$$
   Therefore, we have

   $${\int }_0^{\rho }{\text{cap}}_F\left(\left\{0\right\},B_{2t}(0)\right)\frac{\text{d}t}{t}=c{\int }_0^{\rho }\frac{\text{d}t}{t}=\infty .$$

   In other words, Ω^c is F-thick at 0.

2. On the other hand, since ${\alpha }^{*}(\tilde{F})>0$, we know that a single point is of capacity zero. Therefore, we have

   $${\int }_0^{\rho }\frac{{\text{cap}}_{\tilde{F}}\left(\left\{0\right\},B_{2t}(0)\right)}{{\text{cap}}_{\tilde{F}}\left(\overline{B_t(0)},B_{2t}(0)\right)}\frac{\text{d}t}{t}=0.$$

   In other words, Ω^c is not $\tilde{F}$-thick at 0 and we cannot apply our Wiener’s criterion.

3. Let f₁ ∈ C(∂Ω) is a boundarye data given by

   $$f_1(x)=\{\begin{array}{ll}1\hfill & \text{if }x=0,\hfill \\ 0\hfill & \text{if }\left|x\right|=1.\hfill \end{array}$$
   Then clearly the function $u(x)=1-|x{|}^{1-\frac{\lambda }{\Lambda }}=1-V(x)$ is the solution for this Dirichlet problem. In particular, in this case, we have ${\overline{H}}_{f_1}={\underset{\_}{H}}_{f_1}$ (i.e. f₁ is resolutive) and

   $$\underset{\Omega \ni x\to 0}{\text{lim}}{H}_{f_1}(x)=1=f_1(0).$$

   Alternatively, one can apply Corollary 4.6 to reach the same conclusion, since f₁ attains its maximum at 0 and Ω^c is F-thick at 0.

4. Let f₂ ∈ C(∂Ω) is a boundary data given by

   $$f_2(x)=\{\begin{array}{ll}-1\hfill & \text{if }x=0,\hfill \\ 0\hfill & \text{if }\left|x\right|=1.\hfill \end{array}$$

   Then since the zero function belongs to ${𝒰}_{f_2}$, we have ${\overline{H}}_{f_2}\le 0$. Moreover, since $\epsilon \left(1-|x{|}^{-\left(\frac{\Lambda }{\lambda }-1\right)}\right)\in {ℒ}_{f_2}$ for any ε > 0, we have ${\underset{\_}{H}}_{f_2}\ge \epsilon \left(1-|x{|}^{-\left(\frac{\Lambda }{\lambda }-1\right)}\right)$. Letting ε → 0, we conclude ${\underset{\_}{H}}_{f_2}\ge 0$.

   Therefore, we deduce that ${\overline{H}}_{f_2}={\underset{\_}{H}}_{f_2}=0$. Furthermore, it follows that

   $$\underset{\Omega \ni x\to 0}{\text{lim}}{H}_{f_2}(x)=0\ne -1=f_2(0),$$

   which implies that 0 is an irregular boundary point for Ω.

5. A necessary condition for the regularity of a boundary point

In this section, we provide the necessity of the Wiener criterion, under additional structure on the operator F. Indeed, our strategy is to employ the argument made in [29] which proved the necessity of the p-Wiener criterion for p-Laplacian operator with p > n − 1. Since the assumption p > n − 1 was essentially imposed to ensure the capacity of a line segment is non-zero in [29], we begin with finding the corresponding assumptions in the fully nonlinear case.

Lemma 5.1. Suppose that F is convex and α*(F) > s for some s > 0. Let K be a compact subset in $B_r\left(\subset {ℝ}^n\right)$ such that ℋ^s(K) < ∞, where ℋ^s is the s-dimensional Hausdorff measure. Then

$${\text{cap}}_F(K)=0.$$

Proof. For any δ > 0, define

$${\mathscr{H}}_{\delta }^s(K)≔\text{inf\hspace{0.17em}}\sum _i{r}_i^s,$$

where the infimum is taken over all countable covers of K by balls B_i with diameter r_i not exceeding δ. Then since ${\text{sup\hspace{0.17em}}}_{\delta >0}{\mathscr{H}}_{\delta }^s(K)={\text{lim}}_{\delta \to 0}{\mathscr{H}}_{\delta }^s(K)={\mathscr{H}}^s(K)<\infty$ and K is compact, for each δ ∈ (0, r), there exist finitely many open balls ${\left\{B_i=B_{r_i}\left(x_i\right)\right\}}_{i=1}^N$ such that r_i < δ, ${\cup }_{i=1}^N{B}_i\supset K$, and

$$\sum _{i=1}^N{r}_i^s\le {\mathscr{H}}^s(K)+1<\infty .$$ (5.1)

Now we consider the homogeneous solution $V(x)=|x{|}^{-{\alpha }^{*}}V\left(\frac{x}{\left|x\right|}\right)$ of F. Here we may assume min_|x|=1 V (x) = 1 by normalizing V. If we let $W_i(x)≔r_i^{{\alpha }^{*}}V\left(x-x_i\right)$, then it immediately follows that W_i is non-negative and F-superharmonic in ${ℝ}^n$, and W_i(x) ≥ 1 on B_i.

Finally, we let $W≔{\sum }_{i=1}^N{W}_i(\ge 0)$. Since F is convex, W is F-superharmonic in ${ℝ}^n$. Moreover, W ≥ 1 on ${\cup }_{i=1}^N{B}_i$ and in particular, W ≥ 1 on K. Therefore, $W\in {\Phi }_K^1\left(B_{4r}\right)$ and so

$${\text{cap}}_F\left(K,B_{4r}\right)\le W\left(y_0\right)\le r^{-{\alpha }^{*}}\underset{\left|x\right|=1}{\text{max\hspace{0.17em}}}V(x)\cdot \sum _{i=1}^N{r}_i^{{\alpha }^{*}}\le r^{-{\alpha }^{*}}\underset{\left|x\right|=1}{\text{max\hspace{0.17em}}}V(x)\cdot \left({\mathscr{H}}^s(K)+1\right){\delta }^{{\alpha }^{*}-s},$$

where we used (5.1) and α* > s. Letting δ → 0, we finish the proof. □

Now we prove the partial converse statement of Lemma 5.1. Indeed, here we only consider the compact set K is given by a line segment L, whose Hausdorff dimension is exactly 1.

Lemma 5.2. Suppose that F is concave and α*(F) < 1. Let L = {x₀ +se : ar ≤ s ≤ br} be a line segment in B_r(x₀), where e is an unit vector in ${ℝ}^n$ and 0 < a < b < 1 are constants satisfying $b-a<\frac{1}{2}$. Then

$${\text{cap}}_F\left(L,B_{2r}\right)>0.$$

Proof. Note that since L is a line segment, for any δ > 0, one can cover L by open balls B_i = B_3δ(x_i), 1 ≤ i ≤ N(δ) where x_i ∈ L, |x_i − x_j| ≥ 2δ whenever i ≠ j, and $N(\delta )~\frac{(b-a)r}{\delta }$. We write such cover by $K_{\delta }≔{\cup }_{i=1}^{N(\delta )}\overline{B_i}$. Recalling Lemma 3.9 and its proof, for any ε > 0, there exist a sufficiently small δ > 0 and corresponding cover K_δ such that

$${\text{cap}}_F\left(K_{\delta },B_{2r}\right)\le {\text{cap}}_F\left(L,B_{2r}\right)+\epsilon .$$

If we denote ${\tilde{B}}_i≔B_{\delta }\left(x_i\right)$ and ${\tilde{K}}_{\delta }={\cup }_{i=1}^{N(\delta )}\overline{\widetilde{B_i}}$, then we have ${\tilde{B}}_i$ are pairwise disjoint and

$${\text{cap}}_F\left({\tilde{K}}_{\delta },B_{2r}\right)\le {\text{cap}}_F\left(L,B_{2r}\right)+\epsilon .$$

On the other hand, for simplicity, we suppose that the homogeneous solution V is given by

$$V(x)=|x{|}^{-{\alpha }^{*}}$$

and α*(F) ∈ (0, 1). Note that if α* < 0, then a single point has a positive capacity (Lemma 3.12) and the result immediately follows. Other cases can be shown by similar argument as in Lemma 5.1. For each i = 1, 2, ⋯ , N(δ), write

$$W_i(x)≔{\left(\frac{\left|x-x_i\right|}{\delta }\right)}^{-{\alpha }^{*}}\text{ and }W(x)=\sum _{i=1}^{N(\delta )}{W}_i(x).$$

Since F is concave, W is F-subharmonic in ${ℝ}^n\backslash {\cup }_{i=1}^{N(\delta )}\left\{x_i\right\}$.

1. (On $\partial {\tilde{K}}_{\delta }$) For $y\in \partial {\tilde{K}}_{\delta }$, let y ∈ ∂B_i for some i. Then for j ≠ i, we have

   $$\left|y-x_j\right|\ge \left|x_i-x_j\right|-\left|y-x_i\right|=\left|x_i-x_j\right|-\delta ,$$

   and so

   $$W(y)\le 2\left(1+2^{-{\alpha }^{*}}+\cdots +N{\left(\delta \right)}^{-{\alpha }^{*}}\right)\le 2\left(1+{\int }_2^{N(\delta )}\frac{1}{s^{{\alpha }^{*}}} \text{d}s\right)\le cN{\left(\delta \right)}^{1-{\alpha }^{*}}.$$

   Here we used the condition α* < 1.

2. (On ∂B_2r) For z ∈ ∂B_2r,

   $$\left|z-x_i\right|\ge 2r-br=(2-b)r,$$

   and so

   $$W(z)\le {\left(\frac{(2-b)r}{\delta }\right)}^{-{\alpha }^{*}}\cdot N(\delta ).$$

Therefore, for

$$\tilde{W}(x)≔\frac{W(x)-{\left(\frac{(2-b)r}{\delta }\right)}^{-{\alpha }^{*}}\cdot N(\delta )}{cN{(\delta )}^{1-{\alpha }^{*}}},$$

we have

$$\tilde{W}\text{is }F\text{-subharmonic in }B\backslash {\tilde{K}}_{\delta },\tilde{W}\le 0\text{ on }\partial B_{2r}\text{, and }\tilde{W}\le 1\text{ on }\partial {\tilde{K}}_{\delta }\text{.}$$

Note that since ${\tilde{K}}_{\delta }$ and B_2r are regular domains, the capacity potential ${\widehat{R}}_{{\tilde{K}}_{\delta }}^1\left(B_{2r}\right)$ satisfies:

$${\widehat{R}}_{{\tilde{K}}_{\delta }}^1\left(B_{2r}\right)=0\text{ on }\partial B_{2r},\text{ and }{\widehat{R}}_{{\tilde{K}}_{\delta }}^1\left(B_{2r}\right)=1\text{ on }\partial {\tilde{K}}_{\delta }.$$

Hence, the comparison principle yields that

$${\widehat{R}}_{{\tilde{K}}_{\delta }}^1\left(B_{2r}\right)\ge \tilde{W}\text{ in }{B}_{2r}\backslash {\tilde{K}}_{\delta }.$$

In particular, putting $x=x_0+\frac{3}{2}re$, we conclude that

$$\left|x-x_i\right|\le 3r/2-ar=\left(\frac{3}{2}-a\right)r,$$

and so

$${\widehat{R}}_{{\tilde{K}}_{\delta }}^1\left(B_{2r}\right)\left(x_0+\frac{3}{2}re\right)\ge \tilde{W}\left(x_0+\frac{3}{2}re\right)\ge \frac{\left[{\left(\frac{(3/2-a)r}{\delta }\right)}^{-{\alpha }^{*}}-{\left(\frac{(2-b)r}{\delta }\right)}^{-{\alpha }^{*}}\right]\cdot N(\delta )}{cN{(\delta )}^{1-{\alpha }^{*}}}\ge c_1{\left(b-a\right)}^{{\alpha }^{*}}\left[{\left(\frac{3}{2}-a\right)}^{-{\alpha }^{*}}-{(2-b)}^{-{\alpha }^{-*}}\right].$$

Finally, by applying Harnack inequality for ${\widehat{R}}_{{\tilde{K}}_{\delta }}^1\left(B_{2r}\right)$ on ∂B_3r/2, we have

$$\epsilon +{\text{cap}}_F\left(L,B_{2r}\right)\ge {\text{cap}}_F\left({\tilde{K}}_{\delta },B_{2r}\right)\ge c_2{(b-a)}^{{\alpha }^{*}}\left[{\left(\frac{3}{2}-a\right)}^{-{\alpha }^{*}}-{(2-b)}^{-{\alpha }^{-*}}\right]>0.$$

Since ε > 0 is arbitrary, we finish the proof. □

The idea of the previous lemma can be modified to derive the ‘spherical symmetrization’ result:

Lemma 5.3 (Spherical symmetrization). Suppose that F is concave and α*(F) < 1. Let K be a compact subset in B_r(x₀) such that K meets $S(t)≔\left\{x\in {ℝ}^n:\left|x-x_0\right|=t\right\}$ for all t ∈ (ar, br), where 0 < a < b < 1 are constants satisfying $b<\frac{1}{4}$. Then there exists a constant c = c(n, F, a, b) such that

$${\text{cap}}_F\left(K,B_{2r}\right)\ge c(n,F,a,b)>0.$$

Proof. The proof is similar to the one of Lemma 5.2. By assumption, we can choose x_(t) ∈ K∩S(t) for all t ∈ (ar, br). In particular, for small δ > 0, we define x_i ≔ x_(ar+2δi) for i = 1, 2, ⋯ , N(δ) so that

$$ar+2\delta \cdot N(\delta )<br\le ar+2\delta \cdot (N(\delta )+1).$$

Note that $N(\delta )~\frac{(b-a)r}{\delta }$. Moreover, for δ > 0, we define a set K_δ by

$$K_{\delta }=\underset{i=1}{\overset{N(\delta )}{\cup }}\overline{B_i},$$

where $B_i=B_{x_i}(\delta )$. Again recalling Lemma 3.9 and its proof, for any ε > 0, there exists a sufficiently small δ > 0 such that

$${\text{cap}}_F\left(K_{\delta },B_{2r}\right)\le {\text{cap}}_F\left(K,B_{2r}\right)+\epsilon .$$

On the other hand, for simplicity, we suppose that the homogeneous solution V is given by

$$V(x)=|x{|}^{-{\alpha }^{*}}$$

and α*(F) ∈ (0, 1). For each i = 1, 2, ⋯ , N(δ), write

$$W_i(x)≔{\left(\frac{\left|x-x_i\right|}{\delta }\right)}^{-{\alpha }^{*}}\text{ and }W(x)=\sum _{i=1}^{N(\delta )}{W}_i(x).$$

Since F is concave, W is F-subharmonic in ${ℝ}^n\backslash {\cup }_{i=1}^{N(\delta )}\left\{x_i\right\}$.

1. (On ∂K_δ) For y ∈ ∂K_δ, let y ∈ ∂B_i for some i. Then for j ≠ i, we have

   $$\left|y-x_j\right|\ge \left|x_i-x_j\right|-\left|y-x_i\right|=\left|x_i-x_j\right|-\delta \ge 2\left|i-j\right|\delta -\delta ,$$

   and so

   $$W(y)\le 2\left(1+2^{-{\alpha }^{*}}+\cdots +N{\left(\delta \right)}^{-{\alpha }^{*}}\right)\le 2\left(1+{\int }_2^{N(\delta )}\frac{1}{s^{{\alpha }^{*}}} \text{d}s\right)\le cN{\left(\delta \right)}^{1-{\alpha }^{*}}.$$

   Here we used the condition α* < 1.

2. (On ∂B_2r) For z ∈ ∂B_2r,

   $$\left|z-x_i\right|\ge 2r-br=(2-b)r,$$

   and so

   $$W(z)\le {\left(\frac{(2-b)r}{\delta }\right)}^{-{\alpha }^{*}}\cdot N(\delta ).$$

Therefore, for

$$\tilde{W}(x)≔\frac{W(x)-{\left(\frac{(2-b)r}{\delta }\right)}^{-{\alpha }^{*}}\cdot N(\delta )}{cN{(\delta )}^{1-{\alpha }^{*}}},$$

we have

$$\tilde{W}\text{ is }F\text{-subharmonic in }B\backslash K_{\delta },\tilde{W}\le 0\text{ on }\partial B_{2r},\text{ and }\tilde{W}\le 1\text{ on }\partial K_{\delta }\text{.}$$

Note that since K_δ and B_2r is regular domains, the capacity potential ${\widehat{R}}_{K_{\delta }}^1\left(B_{2r}\right)$ satisfies:

$${\widehat{R}}_{K_{\delta }}^1\left(B_{2r}\right)=0\text{ on }\partial B_{2r},\text{ and }{\widehat{R}}_{K_{\delta }}^1\left(B_{2r}\right)=1\text{ on }\partial K_{\delta }\text{.}$$

Hence, the comparison principle yields that

$${\widehat{R}}_{K_{\delta }}^1\left(B_{2r}\right)\ge \tilde{W}\text{ in }{B}_{2r}\backslash K_{\delta }$$

In particular, putting $x=x_0+\frac{3}{2}r{e}_1$, we conclude that

$$\left|x-x_i\right|\le 3r/2+br=\left(\frac{3}{2}+b\right)r,$$

and so

$${\widehat{R}}_{K_{\delta }}^1\left(B_{2r}\right)\left(x_0+\frac{3}{2}r{e}_1\right)\ge \tilde{W}\left(x_0+\frac{3}{2}r{e}_1\right)\ge \frac{\left[{\left(\frac{(3/2+b)r}{\delta }\right)}^{-{\alpha }^{*}}-{\left(\frac{(2-b)r}{\delta }\right)}^{-{\alpha }^{*}}\right]\cdot N(\delta )}{cN{(\delta )}^{1-{\alpha }^{*}}}\ge c_1{\left(b-a\right)}^{{\alpha }^{*}}\left[{\left(\frac{3}{2}+b\right)}^{-{\alpha }^{*}}-{(2-b)}^{-{\alpha }^{-*}}\right].$$

Hence,

$$\epsilon +{\text{cap}}_F\left(K,B_{2r}\right)\ge {\text{cap}}_F\left(K_{\delta },B_{2r}\right)\ge c_2{\left(b-a\right)}^{{\alpha }^{*}}\left[{\left(\frac{3}{2}+b\right)}^{-{\alpha }^{*}}-{(2-b)}^{-{\alpha }^{*}}\right]>0.$$

Since ε > 0 is arbitrary, we finish the proof. □

Let E be a regular set in a ball B_2r. Let $u={\widehat{R}}_E^1\left(B_{2r}\right)$ be the capacity potential. For γ ∈ (0, 1), let

$$A_{\gamma }=\left\{x\in B_{2r}:u(x)<\gamma \right\}.$$

Lemma 5.4. Suppose that F is concave and α*(F) < 1. Then, there exists a constant c₁ > 0 depending only on n, λ, Λ such that: if

$$\gamma \ge c_1{\text{cap}}_F\left(E,B_{2r}\right),$$

then the set A_γ contains a sphere $S(t)≔\left\{x\in {ℝ}^n:\left|x-x_0\right|=t\right\}$ for some t ∈ (r/10, r/5).

Proof. For 0 < γ < 1, let E_γ ≔ {x ∈ B_2r : u(x) ≥ γ}. We argue by contradiction: suppose that A_γ does not contain any S(t) for t ∈ (r/10, r/5). Then the set E_γ meets S(t) for all t ∈ (r/10, r/5) and we have

$${\text{cap}}_F\left(E_{\gamma },B_{2r}\right)\ge c(n,F)>0,$$

by employing Lemma 5.3 for a = 1/10 and b = 1/5.

On the other hand, by Lemma 4.3, we have

$${\text{cap}}_F\left(E_{\gamma },B_{2r}\right)=\frac{1}{\gamma }{\text{cap}}_F\left(E,B_{2r}\right).$$

Combining two estimates above, we obtain

$$\gamma \le \frac{1}{c(n,F)}{\text{cap}}_F\left(E,B_{2r}\right).$$

Therefore, by choosing $c_1=\frac{1}{c(n,F)}+1$, we arrive at a contradiction. □

Now we are ready to prove the necessity of the Wiener criterion, Theorem 1.3.

Proof of Theorem 1.3. For simplicity, we write B_r = B_r(x₀). Suppose that Ω^c is not F-thick at x₀ ∈ ∂Ω, i.e.

$${\int }_0^1{\text{cap}}_F\left(\overline{B_t}\backslash \Omega ,B_{2t}\right)\frac{\text{d}t}{t}<\infty .$$

For ε > 0 to be determined, choose r₁ > 0 small enough so that

$${\int }_0^{r_1}{\text{cap}}_F\left(\overline{B_t}\backslash \Omega ,B_{2t}\right)\frac{\text{d}t}{t}<\epsilon .$$

Set r_i+1 = r_i/2 and

$$a_i={\text{cap}}_F\left(\overline{B_{r_i}}\backslash \Omega ,B_{2r_i}\right).$$

Applying Lemma 3.14,

$$\sum _{i=2}^{\infty }{a}_i\le c_0(n,\lambda ,\Lambda )\epsilon .$$

Next, by Lemma 2.27 and Lemma 3.9, for each i, choose a regular domain E_i such that $\overline{B_{r_i}}\backslash \Omega \subset E_i$ and

$$b_i≔{\text{cap}}_F\left(E_i,B_{2r_i}\right)<a_i+\epsilon \cdot 2^{-i}.$$

Then we have

$$\sum _{i=2}^{\infty }{b}_i\le \left(c_0+1\right)\epsilon$$

and so b_i ≤ (c₀ +1)ε for i = 2, 3, ⋯ . Moreover, let $u_i≔{\widehat{R}}_{E_i}^1\left(B_{2r_i}\right)$ be the capacity potential. By Lemma 5.4, for γ_i = c₁ · b_i, the set

$$A_i=\left\{x\in B_{2r_i}:u_i(x)<{\gamma }_i\right\}$$

contains S(t_i) for some t_i ∈ (r_i/10, r_i/5). Now by selecting $\epsilon =\frac{1}{2\left(c_0+1\right)c_1}>0$, we have γ_i < 1. In particular, since u₂ = 1 on E₂ and S(t₂) ⊂ A₂, we conclude that S(t₂) ⊂ Ω.

Next, let f ∈ C(∂Ω) be the boundary function defined by

$$f(x)=\{\begin{array}{ll}1\hfill & \text{if }x\in B_{t_2}\cap \partial \Omega ,\hfill \\ 0\hfill & \text{if }x\in \partial \Omega \backslash B_{t_2}.\hfill \end{array}$$

Then we have the following results for the lower Perron solution ${\underset{\_}{H}}_f={\underset{\_}{H}}_f(\Omega )$:

1. ${\underset{\_}{H}}_f\cancel{\equiv }1$: Choose r > 0 large enough so that Ω ⊂ B_r. Moreover, set a domain ${\Omega }_0≔B_r\backslash \left(B_{t_2}\cap \Omega \right)$ and a boundary function f₀ ∈ C(∂Ω₀) by

   $$f_0(x)=\{\begin{array}{ll}1\hfill & \text{if }x\in B_{t_2}\cap \partial \Omega ,\hfill \\ 0\hfill & \text{if }x\in \partial B_r.\hfill \end{array}$$

   Then since B_r is regular, we have ${\overline{H}}_{f_0}\left({\Omega }_0\right)<1$ in $B_r\backslash B_{t_2}$. On the other hand, for any v ∈ ℒ_f(Ω) and $w\in {𝒰}_{f_0}\left({\Omega }_0\right)$, one can check that v ≤ w in Ω using the comparison principle.

   Therefore, we conclude that ${\underset{\_}{H}}_f(\Omega )\le {\overline{H}}_{f_0}\left({\Omega }_0\right)$ and so ${\underset{\_}{H}}_f(\Omega )\cancel{\equiv }1$.

2. ${\text{max\hspace{0.17em}}}_{S\left(t_2\right)}{\underset{\_}{H}}_f=:M<1$: This is an immediate consequence of the strong maximum principle for ${\underset{\_}{H}}_f$ and part (i).

For $u≔\frac{{\underset{\_}{H}}_f-M}{1-M}$ which is F-harmonic in Ω and u ≤ 0 in S(t₂), we claim that

$$\underset{\Omega \ni x\to x_0}{\text{lim inf }}u(x)<\frac{1}{2}.$$ (5.2)

Indeed, since $S\left(t_2\right)\subset B_{r_3}$ and E₃ is a regular domain, we have

$$u(x)\le 0\le \underset{y\to x}{\text{lim inf }}{u}_3(y)\text{ for any }x\in \partial B_{t_2}=S\left(t_2\right),$$

$$\underset{y\to x}{\text{lim sup }}u(y)\le 1=\underset{y\to x}{\text{lim inf }}{u}_3(y)\text{ for any }x\in \partial E_3.$$

Thus, the comparison principle yields that u ≤ u₃ in $B_{t_2}\backslash E_3$. In particular, since S(t₃) ⊂ A₃, we observe that

$$u\le u_3<{\gamma }_3\text{ on }S\left(t_3\right).$$

Iterating this argument (for example, consider u − γ₃ instead of u), we conclude that

$$u\le \sum _{k=3}^i{\gamma }_k\le \sum _{k=3}^{\infty }{\gamma }_k=c_1\cdot \sum _{k=3}^{\infty }{b}_i\le c_1\left(c_0+1\right)\epsilon =\frac{1}{2}\text{ on each }S\left(t_i\right),$$

which leads to (5.2).

Finally, recalling the definition of u, the estimate (5.2) is equivalent to

$$\underset{\Omega \ni x\to x_0}{\text{lim inf }}{\underset{\_}{H}}_f(x)<1=f\left(x_0\right),$$

which implies that x₀ ∈ ∂Ω is an irregular boundary point. □