Regularity of free boundaries a heuristic retro

Abstract

This survey concerns regularity theory of a few free boundary problems that have been developed in the past half a century. Our intention is to bring up different ideas and techniques that constitute the fundamentals of the theory. We shall discuss four different problems, where approaches are somewhat different in each case. Nevertheless, these problems can be divided into two groups: (i) obstacle and thin obstacle problem; (ii) minimal surfaces, and cavitation flow of a perfect fluid. In each case, we shall only discuss the methodology and approaches, giving basic ideas and tools that have been specifically designed and tailored for that particular problem. The survey is kept at a heuristic level with mainly geometric interpretation of the techniques and situations in hand.

1. Introduction

(a). Background

Free boundary problems arise in diverse areas of partial differential equations. As Hans Lewy defined them, they occur when the behaviour of the variables changes discontinuously across one of its values: (i) the solid–liquid interphase when the temperature passes a threshold value for the solidification of some material, (ii) the boundary between exercise and continuation region for a financial instrument (such as option), (iii) the transition from elastic to plastic behaviour when stress value passes a critical level. Because of such applications, free boundary problems are often referred to as phase transition problems.

Mathematically, the problem consists of several things:

• — reconstruction of the domain, where the process takes place,

• — the appropriate boundary data,

• — the interior behaviour laws,

• — shape or evolution of the location of the interphase and

• — the values of the unknown (a density, a temperature, an expectation) on either side of the interphase.

This reconstructing occurs at several levels; e.g., in the melting problem, one first finds a weak formulation, in an appropriate functional space, that allows for the construction of a weak solution, i.e. functions with some integrability properties, stability and a setting that is conducive to numerical simulation. On the other hand, when melting or solidification takes place, we expect the interface to be smooth, except at occasional times when there is total melting, or special configurations.

In that sense, a minimal surface (or movement by mean curvature) is the simplest case of phase transition (the Allen–Cahn model for phase transition). The original model has been scaled in such a way that there is no bulk equation left on the sides of the surface. Indeed, the scaled model neglects what actually happens in the bulk, and becomes solely reflected on the balance among the curvatures of the transition surface.

The theory of minimal surfaces, of course, was developed earlier than most free boundary problems. A particular case, pertinent to our discussion, is the theory of sets of minimal perimeter, i.e. sets where some part of the boundary is an area minimizing surface. The approach, developed by De Giorgi, has important components such as renormalization, blow-up, classification of global solutions, improvement of flatness (these will be explained in §1b).

Two other problems that have related techniques, but where one has to understand the interaction between the interphase and the bulk either sides of it, are the obstacle problem and the cavitation problem. The obstacle problem is a one-phase problem: the bulk is non-trivial only on one side of the transition. A nice visualization is that of a balloon pressed on a table. Vertical downward pressure is exerted on a balloon, which is pressed against the table. The interphase is the separation curve between the contact and non-contact sets. A weak formulation can be given and a solution can be found for instance as a set of finite perimeter.

The cavitation problem appears often as a two-phase problem; in this note, we shall only treat the one-phase case of it. In the stationary case, our unknown u (a stream function or the temperature in flame propagation) takes positive and negative values. Along the zero lever surface, representing an interphase such as the edge of cavity or flame, the normal derivative of u (tangential velocity for cavitation, or combustion energy at ignition for flame) satisfies a jump condition.

A further problem we will discuss is the so-called thin obstacle problem, where the underlying diffusion process is of non-local character, represented by a stable process. This arises in the theory of semipermeable membranes or diffusions with jump processes.

(b). General methodology

In this section, we shall present a road map for the approach to free boundary regularity, based on De Giorgi's ideas for the regularity of boundaries of sets with minimal perimeter, as mentioned above.

Suppose we want to wrap a body, for instance in the shape of a dumbbell, with a minimal amount of a given material. The inside of the wrap will be a set D, with a boundary that partly touches the dumbbell. This gives rise to a surface (which we call S) with minimal area, and in the sense that if we replace it by another configuration the area will increase (or at least not decrease). De Giorgi's theory shows that this part of the boundary of D is an analytic surface. The theory evolves through the following lines:

• — dilation and blow-up,

• — classification of global solutions,

• — positive density and asymptotic of local solutions,

• — improvement of flatness.

(i). Dilation and blow-up

A minimal surface no matter if we look at it from close or afar is still a minimal surface. Hence, minimal surfaces are invariant under dilations. In particular, if we expect to have a tangent plane to S at a point z, and we make a sequence of dilations, the limit should be a plane. Will this be true for all points of a minimal surface? Help comes from a monotonicity formula that says that a minimal surface ‘disorganizes’ as we expand away from the origin; see §1c for a general discussion on monotonicity functions. Here, however, we recall the mean area integral for minimal surfaces, and the monotonicity property of it

1.1

where n≥2 is the space dimension. This monotonicity implies that E(r) has a limit when r tends to zero or infinity, and if we blow up a minimal surface then

which is constant in t. Here, we have denoted the scaling by S_r={x:z+rx∈S} and its limit, when it exists, by S₀.

(ii). Classification of global solutions

Global minimal surfaces are cones, and in fact planes for n<8, where Simon's cone x₁+x₂+x₃+x₄=x₅+x₆+x₇+x₈ is minimal. This along with dilation and blow-up means that in some weak sense local minimal surfaces are asymptotic to a cone (and to a plane in low dimensions), in the sense that the measure of the symmetric difference between the set and a half space goes to zero. Thus, locally, i.e. restricted to a fixed ball around the origin, (sub)sequences of dilations converge to a plane or a cone in measure.

(iii). Positive density and asymptotic of local solutions

The next step is to reinforce the convergence through a uniform density lemma. More exactly, for z∈S, one proves that B_r(z)∩S^± have measure proportional to rⁿ. Here, S^± denotes the set on each side of S. This transforms convergence in measure to uniform convergence, i.e. asymptotically near the origin, and inside a small B_r, the surface S gets trapped into a flatter and flatter cone of width o(r).

The general philosophy of the above argument is the following: in principle, to draw conclusions on a geometric problem from the classification of blow-up solutions, we need to establish that the blow-up limit has kept the information that we are seeking. In the case of minimal surfaces, we are looking at convergence of sets in measure and we want to draw geometric conclusions on their boundaries. This introduces technical difficulties as large pieces of the boundary may disappear. If we are able to show that sets and complements have uniform density along their boundaries, i.e. the intersection of a ball centred at the boundary has uniform positive measure intersecting the set and its complement, the convergence in measure becomes Hausdorff (distance) convergence and the boundaries converge ‘uniformly’ to the limiting boundary. This allows us to retrieve information on the regularity at the original point from the blow-up configuration at that point.

(iv). Improvement of flatness

To complete the theory, we note that as a minimal surface gets flatter closer it comes to a harmonic function; i.e. once the cone passes a ‘critical flatness’-level, flatness starts to improve geometrically. In particular, if the problem is renormalized, such that B_r transforms to the unit ball B₁, and in the vertical direction the surface is flatter than a critical value, we make a vertical dilation only, to make it of unit hight¹ and the minimal surface becomes very close to the graph of a harmonic function, which has bounded second derivatives and becomes very flat near the origin. More exactly, a flat enough minimal surface is even flatter and indeed a C^1,α-graph, and analyticity follows from classical theory.

(c). An indispensable tool: monotonicity function

Monotonicity functions, like that in (1.1), are associated to a kind of ‘radial entropy’ that increases, for a solution of a diffusive process, as we diverge from a given point. More exactly at a given point we have the less energetic and an equilibrium configuration and as we diverge away from the point solutions (and the surface representing them) become more complex and oscillatory. Examples of such configurations are:

• — a plane or a cone for a minimal surface with the above monotonicity function E(r);

• — a harmonic polynomial for Almgren's monotonicity function;²

• — global profiles for a free boundary problem, e.g. through Alt–Caffarelli–Friedman (ACF)-monotonicity function, see §3.

It should be pointed out that uniqueness for the tangent cone in minimal surfaces was proved by Simon [1], who treated the problem in a way that is reminiscent of convergence for time tending to infinity in an evolution equation.³

For a minimal surface, the ratio E(r,S,z)=E(1,S_r,0) grows from the plane configuration, S₀≈ plane, to a higher oscillatory surface S_r.

Another simple monotonicity formula is given by the fact that if u is harmonic, |∇(u)|² is subharmonic and thus its average in B_r increases with r, i.e. the average energy of a subharmonic function over B_r(z) increases in r. This can also be written as the average of the square of the H^1/2-norm of the trace.

In a similar vein, Almgren's monotonicity formula A(r) (see [2]) the ratio, properly normalized on the sphere of radius r, of the square of the H^1/2-norm of the trace over the square of the L²-norm of the function, i.e for a harmonic function A(r) is the Rayleigh quotient of the H^1/2 norm and it increases as we diverge from a point.

For the Signorini problem, once we look at convex solutions, there are three candidates: the solution, the tangential derivative and the normal derivative. The solution changes sign, so it is not a good choice. Between the normal and tangential derivative, we ask which one is extremal for the half space configuration: the choice is then the normal derivative as the corresponding spherical energy-ratio is indeed minimized by the half space solution. See the discussion following equation (3.5).

In the case of the ACF formula [3] (see equation (3.8)), we have to look first at the two-dimensional formula. By comparing half planes with cones in two dimensions, it becomes clear that the relation should be multiplicative, i.e. we should look at the product of the energy averages. There, all formulae are exact for a couple of half planes. In higher dimensions, one realizes immediately that the formula ceases to be exact for two planes: The Minkowski inequality does not match. The matching coefficients are related to the relation between the ‘spherical’ and ‘normal energy’ for homogeneous solutions and the fundamental solution becomes a natural multiplier (see monotonicity function (3.8)).

(d). Organization of this note

As already mentioned, our goal is to give a short account of the regularity theory for the free boundary of the aforementioned problems. The techniques presented in this note have been developed in the last four centuries and are mainly of geometric nature. Readers, unfamiliar with the techniques of free boundary regularity, might find the survey too much of hand-waving, but obviously it would have been impossible to present all details and tools in such a short paper. Non-experts may find it a good idea to combine the reading of this material with references to two recent texts [2,4]. On the other hand, the expert readers may find this as a mind-mapping for the free boundary regularity. Also, in order to avoid a lengthy reference list, we have chosen to refer (when needed) to the latest references, and also to the aforementioned books and the long list of references therein.

2. Obstacle problem

The well-known obstacle problem appears as a minimizer of the Dirichlet energy

over the admissible class of functions . Here, g,φ∈W^1,2(B₁), with obstacle φ and the boundary value g≥φ. The above minimization problem asks for the smallest super-solution in the space , and over the given obstacle φ.

Alternatively, in the case of non-homogeneous or nonlinear problems, it may appear also as the least super-solution above the obstacle, depending on the problem having divergence or non-divergence structure. In the case of the Laplacian, both definitions coincide and are necessary for the full development of the theory.

Existence of the (unique) solution follows by standard methods such as penalization, minimizing the functional, or super-solution technique. We refer to classical books such as [5–7] for some background.

The optimal regularity of the solution of this problem is easily found to be C^1,1, which can be seen through a simple argument (at least for the Laplacian case) as follows. Consider the function

2.1

Then obviously, by the constraint condition and that φ is C² (an assumption that is needed), we have u_h≥φ(x). One can also adjust the boundary value, so that u_h≥u on the boundary.⁴

Moreover, u_h and u are super-harmonic functions and above φ, and u is the smallest such. Hence, u_h≥u and therefore u has bounded second derivatives from below. This along with super-harmonicity of u gives bound on the second derivatives; see [2,8] and the references therein for a more general approach to the regularity of the solutions.

(a). General approach to regularity theory

To study the regularity of the free boundary, it is more appropriate to subtract the obstacle from the solution, that allows for scaling and blow-up. In this note, we shall mainly explain the theory for the case Δφ=−1, as all essential technical elements are already present in this case. The general case can be treated by modification of the methodology for this case. The particular choice of the obstacle having strictly negative Laplacian depends on the need for the non-degeneracy of the problem, see below for a discussion about non-degeneracy.

With this modification, our new problem has changed to having a new function v, with Laplacian being one in the set of positivity, and the solution being C^1,1 in the whole ball. From here, a general standard approach can now be made to prove regularity of the free boundary at points where the free boundary is a priori ‘nice’. The first observation one makes is that asymptotically (from the set of positivity for the solution) at such nice points, all pure second derivatives are non-negative. This observation makes it possible to show that global solutions must be convex; indeed, the harmonicity of the pure second derivatives, their boundedness (due to C^1,1-regularity of the solution), and non-negativity on the boundary, implies by that the maximum principle can be invoked to conclude the positivity of the pure second derivatives for global solutions.

Once we manage to classify ‘blow-up’ limits, where in this problem reduces to two cases:

• (i) blow-up solutions with fat zero set, are one-dimensional half-spaces, and

• (ii) blow-up solutions with thin zero set are polynomials.

Hence, one may consider a separation of the cases, and a priori assumptions on what type of points we start with. In particular, we divide the free boundary points into two groups: (i) points with ‘fat’ zero (coincidence) set, and (ii) points with asymptotically thin zero sets. The first category will lead to regularity of the free boundary close to such sets, in a more standard way, and one shows that the free boundary is locally a smooth graph close to such points. The thin points can be embedded in lower-dimensional smooth manifolds. In this note, we shall also prove partially new results for the free boundary close to singular points.

(b). Reformulation of the problem

The departing point of the theory was to look at free boundary points where there is a non-degeneracy. More exactly, one considers only points z, where Δφ(z)<0, and that φ is smooth enough around the point z. The smoothness required is a Dini continuity for Δφ. This is actually a slightly larger class than assuming Dini continuity of the second derivatives. In this regard, one can mention the paper of Blank [9] where he proves existence of solutions to the obstacle problem with a free boundary that spirals around a point. The construction of I. Blank depends heavily on the fact that the Δφ is non-Dini.

As mentioned earlier, having settled the optimal C^1,1-regularity of the solution, one can consider the equation

which can be written as

with v=u−φ.

(c). Non-degeneracy

The concavity of φ, or more precisely the strict positivity of −Δφ, at free boundary points of interest, becomes of central importance, in order to have the so-called non-degeneracy. That is, we would like to have that the supremum value of v around the free boundary point behaves quadratically with respect to the distance to the free boundary,

2.2

where z is now a free boundary point, and −Δφ(z)≥1 (say). The proof of this follows a simple comparison argument between w(x)=|x−z|²/2n, and v on B_r(z), with z∈{v>0}. Indeed, if (2.2) fails, then w>v(x)−v(z) on ∂B_r(z), and at the same time Δw=1≤Δ(v−v(z)) (in {v>0}), which by strong comparison, principle implies the contradiction 0=w(z)<v(z)−v(z)=0. Letting z→∂{v>0}, we obtain the desired result.

This non-degeneracy along with optimal quadratic growth forces the quadratic scaling of the function v to ‘survive’, in a blow-up process, so that the limit equation exists and has the same properties as the original equation, an invariance property. More exactly, we have the scaled functions v_r(x)=v(rx)/r², satisfying Δv_r=(−Δφ)(rx), so we need this to be non-zero in the limit, as well as v_r be bounded.

(d). Global solutions

The local analysis of the regularity for the free boundary is heavily dependent of the profile of the global solutions, that appears in a blow-up regime. Hence, an important aspect of the theory is to classify global solutions, and then link them to local solutions. However, as any non-negative degree two homogeneous polynomial p(x) with Δp=1 is a solution to the obstacle problem, it becomes crucial to put a priori assumptions on local solutions in order to avoid such cases. One such assumption is the thickness of the coincidence set Λ(v)={v=0}. Indeed, if the origin is a free boundary point, then one can show that for each ϵ>0, there exists a radius r_ϵ, such that if for any r<r_ϵ one has MD(B₁∩Λ(v_r))>ϵ,⁵ then there is a t>0 such that free boundary ∂{v>0}∩B_tr is the graph of a C¹-function [8].

An important step in such an analysis is a uniform almost-convexity of the (local) free boundary. More exactly, one can show that there is a modulus of continuity τ(d(x)) such that D_eev(x)≥−τ(d(x)), where d(x) denotes the distance from the point x to the free boundary. The proof is either direct using Harnack's inequality along with iteration argument [8], or an indirect compactness argument, and using the convexity of global solutions. The convexity of global solutions, again, follow a simple compactness argument.

The convexity along with the thickness assumption MD(B₁∩Λ(v₀))>ϵ implies that the coincidence set Λ(v₀) is a convex set, with non-void interior. In other words, the free boundary for v₀ is locally Lipschitz. Now the function ∂_ev₀, with e directed towards the non-coincidence set Ω₀={v₀>0}, close to the origin is non-negative (due to convexity of v₀) in Ω₀∩B_r for some small r, and also is harmonic there. In other words, we have a positive harmonic function ∂_ev₀ in B₁∖Λ(v₀), and that Λ(v₀) is convex. Obviously, if Λ(v₀) is not C¹ close to the origin, then v₀ is not Lipschitz,⁶ which contradicts the fact that v₀ is C^1,1.

(e). Local analysis

From the local C¹-regularity of the free boundary for global solutions (as shown above), we know that the free boundary of global solution ∂{v₀>0}∩B_tr (for a universal t>0) is almost flat and that the local solution in a rotated system in B_tr.

The final step now follows by showing that w:=C_rD_ev−v≥0 in some smaller ball B_t0r (for t₀=t₀(e)), where e⋅e₁>0.⁷ This implies that the Lipschitz norm of the free boundary can be made as small as we wish (by taking e⋅e₁≈0), and consequently the free boundary is C¹ at the origin. As minimal diameter property propagates with ϵ/2-value to all free boundaries in B_r/2, we may derive the same argument for all free boundary points in B_t0r/4 and claim that the free boundary is C¹, in B_t0r/8.

(f). Regularity close to singular points

The analysis of the singular points (in higher dimensions) were quite untouched until the results of the first author [8]. The result in [8] states that if z is a singular free boundary point, i.e. any blow-up of u at this point gives a polynomial solution (with ), then the singular points of the free boundary (in a uniform neighbourhood of z) lie in a C¹-manifold of co-dimension k. Here, 1≤k≤n is the number of non-zero a_j, and the neighbourhood depends on the smallest a_j.

We shall now give a different description of the singular free boundaries, by simple directional monotonicity idea expressed above. This approach also applies to many other operators than Laplacian, e.g. fully nonlinear or non-divergence type operators. The approach in [8] applies only to the divergence type operators with Lipschitz ingredients.

Let us now suppose the origin is a singular free boundary point, i.e any blow-up becomes a polynomial of degree two , with . We shall consider two separate situations.

Case 1: , a_i=0 for i≠1:

In this case, we consider the function w:=D_eV −v, with V =D₁v. A simple application of directional monotonicity lemma, as suggested in the local analysis case, also applies here,⁸ and shows that for all directions e=(α₁,…,α_n) with α₁>0 we have w≥0 in B_r/2, for r<r₀ (universal r₀). This in particular means that the ‘extended’ free boundaries Γ^±:=∂{±D₁v>0} are each Lipschitz graphs, with Lipschitz norm zero at the origin (i.e. C¹ at the origin), hence they meet tangentially at the origin. From here it is not hard to show that both Γ^± are C¹-graphs. Actually, one needs more detailed argument, but it is a matter of repeating standard analysis for local regular points, as was done in local regularity theory, with Lipschitz free boundaries for all other non-singular points, close to a singular point. The details are left to the reader, and maybe mimicked by the use of techniques in [2], ch. 8.

Case 2: a_i=0, for i=k+1,…,n, and a_i>0 for i=1,…,k:

Such solutions create a cusp-shape surface in directions x_k+1,…,x_n, close to a singular point. For this case, we consider w:=D_eV −v, with V =D_νv and ν⊥e_i for i=k+1,…,n. A similar reasoning as above gives that the surface ∂{±D_νv>0} are graphs in ν-direction, Lipschitz in e_i-directions for i=k+1,…,n, with the Lipschitz norm at the origin being zero. The details are left to the reader.

3. Thin obstacles

The thin obstacle problem refers to finding the smallest super-solution above a lower-dimensional obstacle, i.e. the obstacle φ is now replaced by , where is an (n−1)-dimensional smooth manifold. Also the boundary values should stay above the obstacle . Applications of this model appears in problems in chemistry and biology (describing the solvent flow through semipermeable membrane), financial mathematics (option pricing with jump process) and many other areas.

Also in this case, for clarity, we consider a very simple model with , and φ≡0. In lay terms, the problem is formulated as

3.1

where Λ(u):={u=0}⊂{x_n=0} is a lower dimensional coincidence set.

The regularity theory of the free boundary, for both classical and thin obstacles, follows the same line of reasoning. However, the optimal regularity of the solution u, for the thin obstacle is substantially harder, and was unsolved until 2005, when a proof was given by Athanasopoulos & Caffarelli [10]. The simple approach in the obstacle case above can also be mimicked here by considering a smoothing of the obstacle φ_ϵ:=(−1/ϵ)|x_n|². For the smooth case, the solution u^ϵ will converge to u in some reasonable space, and at least pointwise. Define the corresponding function as in (2.1), with h=(h₁,…,h_n−1,0). Then similar reasoning as in the previous case along with the fact that φ_ϵ is independent of tangential direction τ⊥e_n implies . The extra term C comes from adjusting the boundary condition with the amount C|h|² in the variation (2.1). Letting ε tend to zero, we conclude

3.2

This almost convexity/concavity can turn into convexity/concavity in a blow-up regime. Indeed, if we consider homogeneous scaling u_r(x):=u(rx)/S_r, where S_r=∥u∥_L²(∂Br), then we see that , and if S_r≫r², then . Similarly, we have . This formally suggests that a blow-up solution must be convex in tangential, and concave in orthogonal direction.

These observations, combined with a monotonicity formula, will constitute the main tools for the proof of the optimal regularity of the solution, which is C^1,1/2 on each side of and up to . The regularity theory of the free boundary, which in this case is an (n−2)-dimensional object, will then follow the same lines of reasoning as that of the classical obstacle problem.

In this section, we shall discuss the regularity theory for the so-called thin obstacle problem, i.e. the obstacle in our previous section lies on an (n−1)-dimensional manifold, here denoted , which divides B₁ into two parts. For given functions and satisfying g>φ on , consider the problem of minimizing the Dirichlet integral

over the closed convex set

The main difference from the classical obstacle problem is that u is constrained to stay above the obstacle φ only on and not on the entire domain B₁.

For clarity of exposition, we shall consider a simplified case here by assuming . One may also restrict the problem to upper half ball with zero thin obstacle on B₁′={x_n=0} (this is known as Signorini problem). As a minimizer is a distributional solution to (3.1), one may consider the problem only in the upper half ball . Both existence and regularity of solutions can then be shown in terms of approximation of the thin obstacle with a ‘thick’ one, or the following penalized problem

3.3

where is such that

Using classical PDE techniques, one can prove C^1,α-regularity of the solution up to the boundary B₁′[11]. We shall not discuss the low-regularity result, since besides the reference [11] one may also find a detailed account of this in [2], where the approach follows classical results as in [12].

From C^1,α regularity of u on , our problem can be reformulated as

3.4

(a). Optimal regularity of solutions

The C^1,1/2 regularity was pointed out in two dimensions by Hans Lewy, using complex variables (see below) and proved in full generality by Richardson [13].

In arbitrary dimensions, this was first proved by Athanasopoulos–Caffarelli [10]. Let us first sketch an argument, that shows heuristically why the optimal order of the growth of the solutions from the free boundary should be of order .

We consider the problem in the whole ball B₁, either in the original setting or by even reflection of the solution in the upper half ball . Doing this, we have actually a three-phase problem, where one phase u₁=u₊ is given by the harmonic extension of u across the Neumann boundary part, and then u₂=u₋ in , and u₃ is the reflection of u₂ in B₁′. In this way, we may recall the three-phase monotonicity formula [14], which states

Here,

where infimum is taken over all three-partitions of the unit sphere Sⁿ⁻¹=ω₁∪ω₂∪ω₃ and .

It is known that for n=2, one has β(2)=3, and in higher dimensions β(n)>2 (see [14], §5). This suggests that if our solution has a growth of order a₀ at the origin, then by the above construction of u_i, we must have |∇u_i|≈r^a₀−1, and hence formally φ(r)≈r^{6(a₀−1)−3β+6}. For the latter to be bounded, one needs 6(a₀−1)−3β+6≥0. As, for n≥3, β(n)>2, we must have 6(a₀−1)+6>6, which gives a₀>1. In dimension two, the exact value β=3 gives us 6(a₀−1)−9+6≥0, which amounts to .⁹

The above formal discussion implies that we are expecting a regularity of order at free boundary points, at least in dimension two. As the problem does not seem to be dimensional dependent, we expect the same order of growth in all dimensions. The astute reader has already noted that the idea above already suggests for a new approach to the optimal regularity of solutions to thin obstacle problem, with possible bearings to other problems, especially the extension problems.

If the monotonicity function above could be more precise with exact value for β, then one would be able to have exact results for the optimal behaviour of the solutions in higher dimensions. In [10], the authors introduce a monotonicity function which is considered only in and uses strongly the almost-convexity properties of the set {u=0}. The result in [10] states that

3.5

where u is the solution to the thin obstacle problem. This monotonicity result uses the fact that the smallest eigenvalue, given by

where ∇_θ denotes the surface gradient on the unit sphere, and is given by λ₀=(2n−3)/4.

Using this monotonicity formula, one can prove the growth rate of order 3/2 from free boundary points. To prove this, it suffices to show that .

As a result of almost convexity of the coincidence set, u_xn vanishes on at least half of the ball along the boundary. Hence, we can invoke Poincare inequality along with (3.5) to arrive at

Next, the subharmonicity of implies

which is the desired result.

(b). Regularity of the thin free boundary

The regularity theory for the free boundary of the thin obstacle problem follows the same lines of proof as that of the standard obstacle problem. In other words, we need to classify global solutions, at least those coming from local blow-ups and are not degenerate. As there is a large class of homogeneous global solutions, u_κ= Re(x₁+i|x₂|)^κ with , or κ=2m with , we have to impose some a priori conditions in order to be able to obtain reasonable results. For example, the function solves our problem, with Λ(u) having void interior, and the free boundary is of higher co-dimension. So far there are not many results for the regularity theory close to points with fast decay that give rise to blow-up solutions with free boundaries that are (n−2) dimensional planes. Indeed, such points are unstable, as there is no natural and inbuilt non-degeneracy in the formulation of the problem, and small perturbation may change the configuration. In this direction, some partial information is implicitly provided by a version of Almgren's monotonicity formula (see §1c) that implies that all blow-up limits are homogeneous of a degree higher than two except for the 3/2 and quadratic polynomials.

In this note, we shall stick to the case of solutions with growth κ=3/2, which we call non-degenerate solutions. Also, we shall only present the general framework, pointing out the main ideas and technical passages that might differ from the proof of the obstacle problem. In the discussion below, we will avoid the use of Almgen's formula by classifying 3/2 blow-up limits independently, through ACF-monotonicity formula to reduce it to two dimensions and then by simple two-dimensional computations related to analytic function theory we deduce a classification.

For regularity of the free boundary, we shall confine ourselves to solutions that do not decay faster than order 3/2, at the origin (our free boundary point). Hence, by the optimal growth result, we are confined to solutions u with

3.6

From here we have that the scaled functions u_r(x)=u(rx)/r^3/2 satisfies (formally) the improved convexity

3.7

We also remark that the constant C in (3.7), as was found in (3.2), depends only on the regularity of the ingredients, and not on u. Therefore, we may allow scaling with varying points, and functions (to keep uniformity) which means that we may now consider global solution (not necessarily homogeneous).

The first observation is that u(x′,0)≢0, since otherwise we have a harmonic function in {x_n>0}, with zero boundary data, and growth of order 3/2. Liouville's theorem then implies that the solution is linear, which in turn contradicts (3.6). From here and the convexity of the coincidence set (for global solution), we must have that the free boundary of global solutions (not only homogeneous global solutions) are Lipschitz graphs.

Now a global solution (with growth 3/2) can be shown to be two-dimensional, by the use of ACF-monotonicity function [2]:

3.8

As u∈C^3/2, we have |∇∂_eu|²≈|x|⁻¹, and

where is the volume of the support of (∂_eu)^±. As u is non-degenerate, we have , and hence ∂_eu≢0 (otherwise the function is one-dimensional, which cannot happen). This means that at least one of the limits is non-zero. Hence,

and consequently ∂_eu does not change sign, for any tangential direction. This in turn (by way of calculus) implies that the global function u depends only on two variables (e,e_n), where e⊥e_n. Moreover, ∂_eu does not change sign, so we may assume e=e₁ and ∂₁u≥0.

We shall now prove that global two-dimensional solutions u(x₁,x₂) are homogeneous and then by simple computations one can prove u(x)=Re(x₁+i|x₂|)^3/2. To see this, we first note that ∂₁u∂₂u is harmonic, has linear growth and is zero on {x₂=0}. Hence, by Liouville's theorem,

3.9

up to a multiplicative constant. Differentiating in x₁, and using the harmonicity of u gives

Integrating in x₂-direction, we have

3.10

As the left-hand side is harmonic (either compute it or use complex function theory), we must have f(x₁) is harmonic, i.e. f(x₁)=c₁x₁+c₂. Now the left-hand side in expression (3.10) being zero at the origin, implies c₂=0. Hence, we arrive at (∂₂u)²−(∂₁u)²=c₁x₁. We may w.l.o.g. choose the constant c₁=−2.¹⁰ Now using equation (3.9), and setting w=(∂₁u)², we have i.e. w=x₁+|x|, or . Hence, u is homogeneous of degree and is given explicitly as above.

Having settled the classification of global solutions under condition (3.6), we can now apply the directional monotonicity formula as was done in the standard obstacle problem.¹¹ More exactly we consider again C_rD_eu−u and claim this function is non-negative in B_r/2, for all directions e=(α₁,…,α_n−1,0) with α₁>0, and r=r_α1. Hence, the free boundary is Lipschitz, with Lipschitz norm being zero at the origin. As this process is independent of the free boundary point and depends mainly on condition (3.6), we conclude the C¹-regularity of the free boundary. The details are left to the reader.

Higher regularity of the free boundary can be shown by slightly elaborated techniques, of improved boundary Harnack, applied to ∂_eu/∂_nu in B₁∖Λ(u) (here, one considers the problem in the whole ball). First, one needs the domain B₁∖Λ(u) to have Lipschitz boundary (which does not, as the coincidence set is lower dimensional). This is easily amended by a by-Lipschitz map to a Lipschitz domain [2], p. 186. For the boundary Harnack, we also need ∂_eu≥0 local close to the origin. This is also true by the directional monotonicity above. Hence, one can conclude that for i=1,…,(n−1), ∂_iu/∂₁u∈C^α(B_r) for some small r, and α>0 with simple calculus one can now derive the C^1,α-regularity of the free boundary.

4. Minimal surfaces

The third object of our interest is a Lipschitz minimal graph, which describes for instance the equilibrium state of a soup film over a domain and with given boundary curve. To see this in practice, a simple experiment can be done by dipping a wire (which is a closed curve, by connecting its two ends) into a soapy water. This gives rise to a membrane spanned by the wire, that minimizes the area functional. If the domain is convex and the graph of the wire smooth, it is not hard to prove Lipschitz regularity of the surface in any dimension. The regularity of such a graph, in any dimension was proved by De Giorgi as part of his solution of the 19th Hilbert problem. We shall present a geometric interpretation of his proof of the regularity of the Lipschitz minimal surfaces.¹²

The two main ingredients of the argument are the following. First, the incremental quotient of Lipschitz minimal surfaces (and as their limit, first derivatives) satisfy a uniformly elliptic equation with bounded measurable coefficients. Although the coefficients depend on the gradient of the solution but the minimal surface being Lipschitz, the coefficients are measurable. If the coefficients (i.e. the gradient) were continuous or close to a constant, the solution (the gradient of u itself) would be Hölder continuous and full regularity would follow. Second, De Giorgi's lemma implies that the solution of a uniformly elliptic equation with bounded measurable coefficients (our incremental quotients or the gradient itself) is Hölder continuous.¹³ This, of course, gives the C^1,α-regularity of the minimal surface. We will basically repeat his technique but in a geometric setting that serves as a guide to treat more complex free boundary problems, such as cavitation problem, where the surface does not, anymore, satisfy a PDE, but interacts instead with the side equations.

To set the scene, for the minimal surface problem, we start by defining what we mean with a minimal surface call it Γ. A minimal surface is a surface that locally minimizes its area, or has mean curvature zero.¹⁴ As a concrete case, we consider the case of sets with locally finite perimeter, i.e. find the set with smallest perimeter among all sets with given boundary surface on ∂B₁, say. Let x_n+1=u(x) represent the minimal surface for , with u Lipschitz function and Lipschitz norm λ₁ in B₁. (This means that the minimal surface is in .) We also recall that u solves a translation invariant, nonlinear uniformly elliptic partial differential equation, and therefore its gradient satisfies the hypothesis of De Giorgi's lemma. These properties are in principle enough to derive higher regularity for the solution, by the De Giorgi, Nash Moser theory.

We next observe that, the minimal surface being a Lipschitz graph amounts to the existence of maximal (truncated) cones , touching the minimal surface Γ∩B_r from the above (respectively, below) at every point of the graph. The angle that the cone forms with the horizontal plane coincides with the Lipschitz norm of the graph, that is also the ‘defect’ of u being a horizontal plane. Our objective is the following: we consider a nested family of cylinders, of size 2^−k with centre at the origin, and we want to show that we can enlarge the cones (tilting slightly the coordinates in each step) as we take smaller cylinders, such that the defect also decreases geometrically. This would be equivalent to showing C^1,α regularity of the solution.

How do we find the direction in which the cone would ‘expand’? Suppose we are in two space dimensions, and our surface is a curve with Lipschitz norm one. This means that we can translate it to the right and upward at 45° or more, and it would slide upward. Alternatively, we can translate it to the left and upward at 45° degrees. The translation to the right, for instance, may stick to the original graph most of the time, say for 90% of the points, but then the translation to the left will separate strictly form the original graph 90% of the time. Then, De Giorgi's lemma says that in B_1/2, the translation to the left will be uniformly away from u. In other words, while on the right we keep the 45° line as the edge of our cone, on the left we can choose a smaller slope: the cone is not vertical anymore, but it is wider. As minimal surfaces and Lipschitz norms are invariant under dilations, we can dilate and repeat the above argument to conclude the proof.

In higher dimensions, we have to follow the cone generatrix by generatrix, this we will discuss in more detail in the text below. This in turn implies that we can slide the graph along a direction ξ, with ξ∈K⁺ (|ξ|=1), according to u_ξ(x):=u(x+δξ) (for δ>0 small), and keeping u_ξ≥u in B_1/2. As the function u(x) represents a minimal surface (and thus u_ξ does so) the strong comparison principle implies u_ξ>u, unless u_ξ≡u in B_1/2; the latter can occur for a plane solution. It is also noteworthy that as our solution is defined in B₁, a slide of the graph would not have much meaning in the entire B₁, and hence we consider the comparison in the half-ball (or alternatively, we assume the minimal surface is considered in B₂).

The above comparison argument can be made quantitative, as done by De Giorgi:

If a dislocation of a small distance δ in direction ξ gives

then everywhere in B_1/2 one must have u_ξ−u>C_μθδ, where C_μ is independent of u.¹⁵

To see how exactly this works, we let λ₁ be the Lipschitz constant for the graph in B_1/2, and we want to prove that this can be made strictly smaller in B_1/4, at least for a cone of directions. Now we cover the unit sphere with symmetric spherical caps/patches of opening π/4 (consider cones of aperture π/4, and its intersection with the unit sphere). We may, in particular, consider a finite cover S_k (k=1,…,m), and then define

Obviously, at least for one k=k₀, we have V olume(A_k0)≥c₀, for some c₀=c₀(m,n). Hence, we can apply De Giorgi's argument to the function w:=D_−eu−λ₁, which solves a linear PDE, with bounded measurable coefficients, and is uniformly elliptic and satisfies w≤0 on A_k0, and conclude w≤−μc₀ in B_1/4.

This implies that our cone is not maximal anymore, and hence there is another cone, which in a uniform set of directions is larger than our cone and with a uniform amount, and touches the minimal surface from the above. Now we can tilt the surface with half of this amount in one of these directions such that the new surface has a maximal cone situated symmetrically in the chosen direction, above the minimal surface. In particular, this means that there are uniform cone of directions in (being opposite side of each other) with vertex at the free boundary, and such that the cones have Lipschitz norm λ₁(1−τ), for some universal 0<τ<1. This is also the case of all points in B_1/2, by invariance of the problem.

Repeating this argument in B_1/2^k, for k=2,3,…, we have a sequence of cones (opposite to each other) and with uniform opening for which the Lipschitz norm diminishes to λ₁(1−τ)^kκ₀, where κ₀<1 is a fixed constant depending on the covering properties of these cones of the unit ball in . This means that the minimal graph is C^1,α at the origin, with . As this property holds for all points, we conclude that the minimal graph is uniformly C^1,α.

5. Free boundaries of cavitation type

The fourth object of interest is the so-called cavitation problem,¹⁶ which amounts to a free boundary problem, where the extra (free boundary) condition is given by a jump of the derivative of the solution function (representing velocity potential). There is also a two-phase version of this problem that we shall not discuss in this note [4]. Several phenomenon in nature lead to mathematical models that are reminiscent of and bear the same characteristics as that of cavitational problem, which is characterized by a jump/discontinuity of the speed across the free boundary. Such problems may appear in phenomenon involving any kind of flow and such that increase in the speed of the flow occurs simultaneously as a decrease in the potential energy of the system.

The mathematical formulation of such problems can be done in several ways: smallest super-solution, singular perturbation and minimization of functionals with discontinuous integrands, are among the models that lead to a free boundary related to cavitation, also referred to as Bernoulli type free boundary. The name is suggested by the extra boundary condition which obeys Bernoulli's law, describing fluid's velocity on the boundary.

The notion of solution we have chosen for this last object of discussion is through a viscosity definition. This amounts to saying that given is a non-negative continuous function u in the unit ball, with the following properties: Δu=0 in {u>0}, and close to a ‘smooth’ boundary point z∈∂{u>0}∩B_1/2 we have an asymptotic development of the type u(x)≈(x−z)₊⋅ν, for some directional vector ν. The question one addresses is: How regular is the free boundary ∂{u>0} close to such points? Naturally, one needs to make this asymptotic behaviour more precise as well as to impose some a priori conditions on the free boundary, or the solution, in order to have a meaningful problem. In this note, we shall depart from Lipschitz free boundaries and derive heuristically the C^1,α regularity of the free boundary.

We start by defining the notion of viscosity (or comparison) solutions¹⁷ for the free boundary problem that we have in mind. As cavitation problems leads to jump in the gradient across the free boundary, one has to start with a reasonable definition of the gradient on the boundary. This is done in the viscosity sense.

Let us first remark two interesting facts about positive harmonic functions:

Let u>0 in D, and harmonic there, along with vanishing boundary values, locally near a boundary point ∂D∩B_r(z).

• (1) If D satisfies the interior ball conditions at z, i.e. B_r1(y)∈D, and z∈∂B_r1(y)∩∂D, then u has asymptotically positive (possibly infinity) slope at z.

• (2) If D satisfies the exterior ball conditions at z, i.e. B_r1(y)∈D^c, and z∈∂B_r1(y)∩∂D^c, then u has bounded Lipschitz norm (possibly zero) at z.

In the light of these properties/facts, one can define ‘forcefully’ a viscosity super- and sub-solution as follows.

(a). Super- and sub-solution

Let u be a positive harmonic function in D, with zero boundary values on ∂D∩B₁.

• (i) We say u is a super-solution at a free boundary point z if whether there is tangential inner ball condition at z, then one has .

• (ii) A sub-solution is defined accordingly by using outer touching ball at z, and with the property that .

• (iii) A viscosity solution is a function that is both super- and sub-solution.

The astute reader may already have noted that this definition is implicitly using the facts (1)–(2) above, concerning the behaviour of positive harmonic functions near boundary points. Indeed, the tangential inner ball condition for super-solutions is using the fact that the gradient is asymptotically non-zero at such free boundaries, and the super-solution condition forces it to stay below one. The latter in turn has a consequence that the free boundary is also smooth from outside.

We shall here consider the case of Lipschitz free boundaries and then derive local C^1,α regularity. The main idea is to use the approach described in the section of minimal surfaces, i.e. improvement of the cone, and Lipschitz norm. The proof of this type of results uses several geometric facts and observations, that we shall now line up here. The observation we can make are the following.

Observation 1: (Regular solutions). The first observation is that is a viscosity sub-solution. For D_ε=∪_x∈DB_ε(x), ∂D_ε is a parallelsurface to ∂D and D_ε has a touching inner ball at every boundary point. It is also apparent that each boundary point z∈∂D_ε realizes its distance to ∂D by touching ball from outside of D to ∂D at some point y_z, i.e. there exists B_r(z)∈D^c, with dist(z,D)=|z−y_z|=r. Hence by sub-solution property for u, we have in B_r(y_z). This in particular implies in B_r(z), i.e. u_ε is a (almost) sub-solution. This is usually called a Regular sub-solution.

Observation 2: (Cone of monotonicity). The Lipschitz property can be replaced by directional monotonicity for u. More exactly, if K(θ,e_n) is the optimal (open) cone of directions inside D, with vertex at the free boundary, axis e_n, and opening θ, then for any direction τ∈K(θ,e_n), we have ∂_τu(x)≥0 in B₁. Indeed, if we sacrifice a small portion of the opening of the cone, i.e. for those directions that are, for instance, in the cone with half opening, the corresponding directional derivatives are positive in a full neighbourhood of the boundary.¹⁸ This, in particular, implies

5.1

with ε depending uniformly on the ingredients.

Observation 3: (Shifting and regularizing). The next geometric idea is to approximate the free boundary with a smooth surface from the within of D. To see how this works, we may consider a new sub-solution

5.2

This is a translation inwards of the regular sub-solution considered in our first observation, in direction τ. Now by (5.1) v_ε≤u on B_1−ε, and therefore by Hopf's lemma, the free boundaries can not touch in any interior point z since then

This is a contradiction, unless v_ε≡u, which in turn would imply that the free boundary for u is smooth (and hence we would have been done).

The core idea is to make Observation 3 quantitative (in a uniform manner). More exactly we want to see if there are any further information in the problem that makes it possible to improve the ϵ fattening, in a smaller ball. In terms of v_ϵ, this means

5.3

for some small (but universal) δ, and with τ as in (5.2). It should be noted that (5.3) implies monotonicity of u in a larger cone in the ball B_1/2, with the geometric implication that K(θ,e_n) can be enlarged into , for a new direction and slightly larger , which can be shown to be , for a universal μ. Therefore, once we prove (5.3), then an iteration (by considering iteratively 2^ku(x/2^k)) will give us a sequence of cone improvements K(θ_k,e_k), in B_2^−k, with θ_k=π/2−μ(θ_k−1−π/2)=⋯=π/2−μ^k(π−θ). This in turn implies C^1,α regularity at the origin. As this holds with same constant in a uniform neighbourhood of the origin, we obtain C^1,α-regularity of the free boundary.

(b). Interior gain on monotonicity

To complete our heuristic proof, we need to prove the implication (5.3). To explain this geometrically, we consider a point z=(3/4)e_n right above the origin. Then the half-space Π₊:={x⋅∇u(z)>0} contains the cone of monotonicity K(θ,e_n). All directional derivatives in directions ν, that go into the half space are of course positive at z, in fact as they go strictly into the half space, away from the edge of Π₊ they become strictly positive at z, proportionally to t(|∇u(z)|, where t is the distance of ν to the edge of Π₊. On the other hand, if the direction happens to be also in the cone of monotonicity, it is non-negative everywhere in our domain, say in a cylinder centred at the origin. Therefore, from Harnack inequality, the directional derivatives that are inside the cone are not only positive but comparable to t|∇u(z)| in, say, half the original cylinder.

Finally, if the direction ν is at the edge of the cone, but strictly inside Π₊, being of size t|ν|, it forces positivity in a full neighbourhood of directions of size t.

If we follow with this gain the edge of K(θ,e_n), we realize that in a ball of size say around z, the cone of monotonicity of u increases its opening proportionally to its defect of being a half space (its axis is generally tilting towards the direction of grad u(z)).

We have won flatness of the level surfaces at least in a region near z.

We now want to transfer that information to the boundary, and we will do so by finding through averages a continuous family of deformation of the level surfaces through sub-solutions all the way to the free boundary.

To that purpose, we point out that a cone is the union of interior tangent balls: If we drop inside a cone, a ball of a given radius will fall until it fits, and the cone is then the union of this family of balls of continuous radii, and becomes a plane when the radius of the ball equals the distance to the origin. At this point, in our inductive step, we have that u has a cone of monotonicity of opening theta is in all of our domain, but near z we have a larger cone of monotonicity. In terms of the inscribed balls that define the cone, we have that at every point x, U(x) is smaller than the infimum of u in the ball of radius r, a uniform distance above x, along the axis of the cone. But we have the extra information that near z, we can enlarge the size of the ball.

We want to show that on a region near the origin, we can get monotonicity in an intermediate cone. If we do that for all radii r, we get in an iterative process, cones whose defect to be a plane decays geometrically to zero, that control by above and below the level surfaces of u. This implies C^1,α regularity of the level surfaces.

(c). Making it work

The analytical tool in proving implication (5.3) relies heavily on the consideration of a point-dependent smoothing. More exactly, one considers

where φ is subject to the condition

5.4

The latter condition implies Δv_εφ≥0, whenever v is harmonic. This in particular implies that we may now consider point-dependent balls of regularization of our shifted function, and hopefully keep the (viscosity) free boundary condition. Hence, the heart of the matter now lies in finding a φ that makes v_εφ a viscosity sub-solution to our free boundary problem. Next, we consider the ring B_7/8(0)∖B_1/8(z) and prescribe for a parameter family of functions φ_t with φ_t=1 on ∂B_7/8, φ_t=1+tμ on B_1/8(z), and satisfying (5.4) in B_7/8(0)∖B_1/8(z). Such a φ_t will take intermediate value between 1 and 1+tμ, on B_1/8(0).¹⁹

We then start with the initial configuration i.e. for t=0. Then the corresponding v is a sub-solution, below u, and we start to deform it continuity by increasing t. For this continuous family to cross u, it has to touch it in the interior because of our choice of boundary data. But, by the comparison principle that cannot happen. That forces them to remain below u and so in a smaller domain we have increased the radius of the balls, proportionally to the defect of the cone of monotonicity being a cone, and since this is valid for any r, we have a new and wider cone of monotonicity.

Competing interests

We declare we have no competing interests.

Funding

L.A.C. was partially supported by NSF grant. H.S. was partially supported by the Swedish Research Council.