Non-Lipschitz minimizers of smooth uniformly convex functionals

Abstract

We construct non-Lipschitz minimizers of smooth, uniformly convex functionals of type I(u) = ∫_Ω f(Du(x))dx. Our method is based on the use of null Lagrangians.

1. Introduction

We consider variational integrals of the form

1.1

where Ω is a bounded open set with smooth boundary in Rⁿ, u: Ω → R^mDu is the gradient matrix of u, and f: M^m×n → R is a smooth uniformly convex function with uniformly bounded second derivatives. Here M^m×n denotes the set of real m × n matrices. [Recall that we say f is uniformly convex if there exists a constant ν > 0 such that for all ξ ∈ M^m×n, X ∈ M^m×n, the inequality (X)ξξ≥ ν|ξ|² holds.]

We shall consider the regularity of minimizers of I belonging to W^1,2(Ω,R^m). By a minimizer we mean a mapping u ∈ W^1,2(Ω, R^m) such that for any smooth mapping φ: Ω → R^m compactly supported in Ω the inequality I(u + φ) ≥ I(u) holds. When f is uniformly convex with uniformly bounded second derivatives, it is not difficult to see that u is a minimizer of I if and only if u is a weak solution of the Euler–Lagrange equation of I, i.e., u is a weak solution of

1.2

(Here and in what follows we use the summation convention.)

A classical result of Morrey (see ref. 1) shows that when n = 2, m ≥ 1, and f is a smooth uniformly convex function with uniformly bounded second derivatives, every weak solution of Eq. 1.2 is smooth; this is also the case when n ≥ 2, m = 1, and f satisfies the same condition by fundamental work of De Giorgi (see ref. 2) and Nash (see ref. 3). The method used in the proof of De Giorgi and Nash cannot be extended to the case m ≥ 2 as shown by a counterexample of De Giorgi (see ref. 4). The first example of a nonsmooth minimizer for a smooth uniformly convex functional of type 1.1 was constructed by Nečas in high dimensions (see ref. 5). He considered u: Rⁿ →Rⁿ² defined by

1.3

and for large n constructed a smooth uniformly convex function f with bounded second derivatives defined on M^n×n2, for which u is a minimizer of the corresponding functional I. Later, Hao et al. (6) were able to modify this construction and make it work for n ≥ 5. They modified the original u in the following way:

1.4

Recently we (see ref. 7) constructed a nonsmooth minimizer of a smooth uniformly convex functional of type 1.1 in the case n = 3, m = 5 by considering the same function u defined by Eq. 1.4. The main idea of our construction is the following. Let K = {∇u(x), x ∈ Ω} be the set of gradients of u. We find a null Lagrangian L (see Definition 2.1) such that

1.5

for a smooth uniformly convex function f with bounded second derivatives. Then u will satisfy the Euler–Lagrange equation of I automatically.

All the counterexamples of nonsmooth minimizers above are Lipschitz-continuous. In fact, it was an open problem whether minimizers with unbounded gradients exist. Partial results in this direction can be found in ref. 8, where local Lipschitz continuity of minimizers for a special class of functionals was obtained.

In this paper we use the null Lagrangian approach to construct counterexamples showing, among other things, that in general for n ≥ 3 we cannot expect Lipschitz continuity of the minimizer of a smooth uniformly convex functional. Moreover, for n = 5 we find a locally unbounded solution to Eq. 1.2. We recall that n = 5 is the first possible dimension where such an example is possible. (When n ≤ 4 each minimizer must be Hölder-continuous, because it belongs to W^2,2+δ for some δ > 0; see ref. 9.) We also construct in section 4 a completely new example for n = 4, m = 3. The important feature in this example is the low dimension of the target space. The construction also gives a non-Lipschitz minimizer in this case. The mapping used in that example is derived from the Hopf fibration S³ → S², which can be thought of as a complex version of Eq. 1.4. In addition, as a byproduct of our methods, we found an example (with n = m = 3) of nonuniqueness of weak solutions of Eq. 1.2 in the spaces W^1,p with 1 < p < 2. This is briefly explained in section 5.

For counterexamples to regularity of solutions of elliptic systems that are not of the form of Eq. 1.2 we refer the reader to refs. 23 and 25. Examples of non-smooth unbounded minimizers of functionals of the form of Eq. 1.1 for integrands with unbounded second derivatives (the so-called p, q-growth conditions) were obtained even in the scaler case in refs. 10–12. A comprehensive treatment of regularity questions can be found in ref. 9. Interesting sufficient conditions for regularity are discussed in ref. 13.

2. Preliminaries

First we introduce some basic facts about null Lagrangians.

Definition 2.1 (see ref. 14):

L: M^m×n → R is a null Lagrangian if for each smooth u: Rⁿ →R^m,

2.1

We recall the following classical theorem about null Lagrangians (see refs. 15 or 16).

Proposition 1.

Let L: M^m×n → R, the following conditions are equivalent:

• (i)  L is a null Lagrangian.

• (ii)  L is a linear combination of subdeterminants.

• (iii)  L is rank-one affine, i.e.t → L(A + tB) is affine for each A ∈ M^m×n and each B ∈ M^m×n with rank B = 1.

Moreover, if L is quadratic, then any of the above conditions are satisfied if and only if L(B) = 0 for each B ∈ M^m×n satisfying rank B = 1.

3. The Case n ≥ 3, m = n(n + 1)/2 − 1

Let Ω be the unit ball in Rⁿ. Consider u^ɛ(x) = (u(x)) given by

3.1

Then for each x ∈ Ω, u^ɛ(x) ∈ {A ∈ M^n×n, A = A^t, TrA = 0} ≅ R^{[n(n+1)/2]−1}. For each R ∈ SO(n) we have

Denote K^ɛ = {∇u^ɛ(x), x ∈ Ω}, K = {∇u^ɛ(x), x ∈ Sⁿ⁻¹}. Following ref. 7, we identify M^m×n with T = {a_ijk ∈ (Rⁿ)^⊗3|a_ijk = a_jik, a_iik = 0} in the obvious way. We recall that m = [n(n + 1)/2] − 1. Then we use a classical procedure to decompose T into irreducible subspaces (see ref. 17). We first decompose T into the trace-free part T′ and its orthogonal supplement T₃, i.e., T = T′ ⊕ T₃. An easy calculation shows that the projection on T₃ is given by a_ijk → −[2/(n + 2)(n − 1)]δ_ijη_k + [n/(n + 2)(n − 1)]δ_kiη_j + [n/(n + 2)(n − 1)]δ_jkη_i with η_k = a_kii, k = 1, … , n. Then we decompose T′ by using symmetrizations. We have T′ = T₁ ⊕ T₂, where the projection on T₁ is given by symmetrizations, i.e., a_ijk → (a_ijk + a_jki + a_kij); the projection on T₂ is given by a_ijk →(a_ijk + a_jik − a_kji − a_kij), which corresponds to the following Young tableau.

We remark that the antisymmetric part of any tensor in T is 0.

By the above formula, a rank one matrix a_ijk = C_ijξ_k in M^m×n with C = C^t, Tr C = 0 can be decomposed as

with

For X = X₁ + X₂ + X₃, X_i ∈ T_i, we let L(X) = −2|X₁|² + |X₂|² + n|X₃|². From the above formula we see that L vanishes on all rank-one matrices in M^m×n, hence L is a quadratic null Lagrangian on M^m×n. Moreover, we have the following lemma.

Lemma 3.1.

We have on K and for

there exists constantδ₀(ɛ) > 0, such that for anyX = ∇u^ɛ(x), Y = ∇u^ɛ(y) ∈ K, we have

3.2

Proof:

First we note that on K we can decompose ∇u^ɛ(x) = {u} as follows:

where u^ɛ,i ∈ T_i with

3.3

and

Hence ∀x ∈ Sⁿ⁻¹,

Because L is quadratic, we have

where we also use L for the symmetric bilinear form corresponding to the quadratic form L.

Let t = 〈x, y〉. Then −1 ≤ t ≤ 1, and we have

Because L(X) = l_ɛ on K, therefore

Note there exist constants c₁(ɛ), c₂(ɛ) > 0, such that for X = ∇u^ɛ(x), Y = ∇u^ɛ(y) ∈ K,

It is clear that when

we can always find δ₀(ɛ) > 0, such that Eq. 3.2 is satisfied.

We proved in ref. 7 that Eq. 3.2 together with the fact that L is constant on K is also sufficient for the existence of a smooth uniformly convex function with bounded second derivatives satisfying Eq. 1.5 on K. We explain the main idea here for the convenience of the reader. A natural attempt to make such an extension would be to take the convex hull of K and consider a modification of the corresponding Minkowski function and then use the homogeneity of L and the Minkowski function. However, because the convex hull of K may not be smooth at K, we need to slightly modify this construction.

We fix μ > 0 (the exact value will be specified later) and for each X ∈ K, consider the ball in T₁ ⊕ T₃ of radius r_μ = μ|∇L(X)| = μm_ɛ passing through X centered at X′ = X − ∇L(X)μ. We will denote the ball as B_X′,rμ.

Lemma 3.2.

When μ is sufficiently small we have

3.4

for each X ∈ K and each Ỹ ∈ B_Y′,rμ, whereB_Y′,rμ is defined above, withY being an arbitrary point of K.

Proof:

The inequality

gives

Hence,

and the statement follows easily.

Let S^ɛ = ∪_X∈KB_X′,rμ. When μ is small, the boundary of S^ɛ is smooth by elementary results about tubular neighborhoods (see refs. 18 or 19). Lemma 3.2 implies that (for sufficiently small μ) all the eigenvalues of the second fundamental form of ∂S^ɛ is negative and bounded above uniformly on K by a negative constant γ_ɛ [i.e. the principle curvatures k_i(X) ≤ γ_ɛ < 0, ∀i and ∀X ∈ K]. Because ∂S^ɛ is smooth, we conclude that ∂S^ɛ is locally strongly convex at any point of U^ɛ ∩ ∂S^ɛ, where U^ɛ is a small neighborhood of K in T₁ ⊕ T₃.

Now take G^ɛ to be the convex hull of S^ɛ in T₁ ⊕ T₃. Using Lemma 3.2 and the fact that ∂S^ɛ is smooth and locally strongly convex in U^ɛ ∩ ∂S^ɛ, we infer that U^ɛ ∩ ∂G^ɛ = U^ɛ ∩ ∂S^ɛ when the neighborhood U^ɛ of K is chosen to be sufficiently small. Let

Then F^ɛ is two-homogeneous, smooth in U^ɛ, and uniformly convex in U^ɛ (see ref. 20), and ∇L(X) = ∇F^ɛ(X) for each X ∈ K.

Let φ be a nonnegative C^∞ function with support in [½, 1] such that

Define

where φ_δ(X) = δ⁻ⁿφ(X/δ). Then F̃ is convex and two-homogeneous (see ref. 21). Let

Let η̃^ɛ(X) be a smooth cutoff function defined on T₁ ⊕ T₃, which vanishes outside U^ɛ and is 1 in a smaller neighborhood V^ɛ of K.

Consider

Define

A direct calculation shows that when δ, τ are small enough, G is two-homogeneous, smooth away from 0, and uniformly convex away from 0 with

3.5

Moreover, by the fact that

we see Eq. 3.5 holds on K^ɛ.

Fix δ, τ such that h^ɛ(X) = G(X) is smooth and strongly convex away from 0. Let ψ(x) be a smooth mollifier defined on R^mn with support in B₁, and let λ^ɛ(X) be a smooth cutoff function defined on R^mn such that λ^ɛ(X) = 1 in B_Nɛ/4 and vanishes outside B_Nɛ/2. Define

where h(X) = h^ɛ ∗ ψ_α(X), ψ_α(X) = α⁻ⁿψ(X/α). It is not difficult to see that when α, β are small enough, f_α,β is smooth and uniformly convex and satisfies Eq. 3.5 on K^ɛ.

In the last step, define

where A = X + Y + Z, X ∈ T₁, Z ∈ T₂, Y ∈ T₃. Then f^ɛ is a smooth, strongly convex function with bounded second derivatives, which satisfies Eq. 1.5 on K^ɛ.

We remark that we can take ɛ > 1 when n ≥ 5 and thus get an unbounded minimizer of a functional of the type 1.1.

4. The Case n = 4, m = 3

In this section, let Ω be the unit ball in R⁴. For ɛ ≥ 0, consider mapping v^ɛ : R⁴ → R³ given by

4.1

where (z, w) ∈ C² ≅R⁴, r² = |z|² + |w|², and ℜf, 𝔍f denote the real and imaginary part of f, respectively.

For R = ( ) ∈ SU(2), |a|² + |b|² = 1, a, b ∈ C, denote by ρ(R) the real four-dimensional representation of SU(2) given by

Where

For R ∈ SU(2), we have

where ρ̃₃(R) is the real representation of SU(2) on ∧²R³ induced by ρ₃(R), the three-dimensional irreducible representation of SU(2). We remark that ρ₃ is of the real type. We have

4.2

where ∇f denotes (∂f/∂x₁, ∂f/∂x₂, ∂f/∂x₃, ∂f/∂x₄).

In another way, we can write Eq. 4.2 as

4.3

for x ∈ R⁴. We then have the following lemma.

Lemma 4.1.

There exists a unique (up to a multiplication by a real scalar) quadratic null Lagrangian onM^4×3, which is invariant on the above action of SU(2).

Proof:

We identify the quadratic null Lagrangians on M^4×3 with ∧²R⁴ ⊗ ∧²R³ ≅ Hom(∧²R⁴, ∧²R³) and consider the representation τ of SU(2) on Hom(∧²R⁴, ∧²R³) induced by ρ̃₃ ⊗ ρ. Using elementary-group representation theory (see ref. 22 or 24), we can easily determine that over the field of complex numbers, τ decomposes into irreducible representations as

where ρ_i denotes the unique i-dimensional irreducible representation of SU(2) over C. Because all ρ_i appearing in this decomposition are of the real type, we can have the same decomposition over the real numbers. The statement follows easily.

A straightforward calculation along standard lines gives the following expressions for the null Lagrangian from Lemma 4.1:

4.4

We recall that the invariant null Lagrangian is defined only up to a multiplication by a real scalar. In what follows we will use the normalization given by Eq. 4.4.

Now we follow the same method used in section 3 to construct the convex function f^ɛ. First we have the following lemma.

Lemma 4.2.

For 0 ≤ ɛ <

− 2, x, y ∈ S³, L(∇v^ɛ(x)) ≡ 2 − ɛ,and there exists constant c₀(ɛ) > 0such that

4.5

Proof:

When r = 1,

Then

4.6

Here we use the same notation for the gradient set of v^ɛ as in section 3.

When x, y ∈ S³, a straightforward calculation yields

Write 〈x, y〉 = t; when t ≥ 0, we have

and when t < 0,

Hence, if we take 0 ≤ ɛ < − 2, the conclusion follows by the fact that

We then can follow the procedure in section 3 to finish the construction of f^ɛ.

5. An Example of Nonuniqueness of Weak Solutions in W^1,2−δ

Let Ω be the unit ball in R³, we consider w^ɛ : Ω → R³ given by

5.1

Direct calculation shows that for L(X) = −Tr cof(X), we have

where K = {∇w^ɛ(x), x ∈ S²}. Then we can follow the same procedure to construct smooth, uniformly convex f^ɛ such that w^ɛ satisfies

5.2

in the sense of distributions. On the other hand, we know u ≡ x is the unique W^1,2 weak solution of Eq. 5.2 from general theory. Note that for our choice of ɛ, w^ɛ is in W^1,p(Ω, R³) for 1 < p < 3/ɛ but not in W^1,2(Ω,R³), which thus gives a counterexample to uniqueness of equations of type 1.2 in W^1,p space.

We summarize what we have proved in Theorem 1 (B denotes the unit ball in Rⁿ).

Theorem 1.

• (i)  Let u^ɛ :B → R^m be given by Eq. 3.1, where m = [n(n + 1)/2] − 1. Then for

  

        there exists a smooth,uniformly convex function f^ɛ : M^m×n → R such that|D²f^ɛ| ≤ c inM^m×n and

  
• (ii)  Let v^ɛ :B → R³ be given by Eq. 4.1. For 0 ≤ ɛ < − 2, there exists a smooth, uniformly convex functionf^ɛ : M^3×4 → Rsuch that |D²f^ɛ| ≤ c in M^3×4 and

  
• (iii)  Let w^ɛ :B → R³ be given by Eq. 5.1. For < ɛ < 3,there exists a smooth, uniformly convex functionf^ɛ : M^3×3 → Rsuch that |D²f^ɛ| ≤ c in M^3×3 and