Overdetermined problems for the normalized -Laplacian

By Agnid Banerjee and Bernd Kawohl

Abstract

We extend the symmetry result of Serrin Reference 21 and Weinberger Reference 24 from the Laplacian operator to the highly degenerate game-theoretic -Laplacian operator and show that viscosity solutions of in , and on can only exist on a bounded domain if is a ball.

1. Introduction

In a seminal paper Reference 21 Serrin showed that the following overdetermined boundary problem can only have a solution if is a ball.

Here is constant and is a bounded connected domain with boundary of class . Serrin used Alexandrov’s moving plane method for his proof, while Weinberger Reference 24 found a proof using Rellich’s identity and the fact that a related function is constant in . Only the second method of proof has been adapted to a situation where the Laplacian operator is replaced by the -Laplacian in Reference 9 and Reference 13.

In this paper we treat the case that the Laplacian is replaced by the normalized or game-theoretic -Laplacian which is defined for any by

a convex combination of the limiting operators

Note that this operator is not in divergence form. Therefore it resists attempts to treat it with variational methods. On the other hand it is quite benign, because its coefficient matrix is bounded from below by and from above by . Therefore the moving plane method seems more appropriate in this context.

Note also, that the above definition of the normalized -Laplacian needs further explanation when . The definition of and a weak comparison principle for continuous viscosity solutions are given below. These and an existence and uniqueness result can be found for instance in Reference 18 or Reference 17. Our main result answers an open problem from Reference 14.

Theorem 1.1.

For let be a viscosity solution to the overdetermined boundary value problem

on a connected bounded domain with boundary of class . Then must be a ball.

Remark 1.2.

We note that the Neumann condition is interpreted in the following sense: Any function such that has a minimum at a point satisfies at . Similarly, any function such that has a maximum at a point satisfies .

Remark 1.3.

It was pointed out in Reference 14 that Theorem 1.1 remains true for , while for it is generally false.

In fact for the equation can be rewritten as , where denotes mean curvature of the level set passing through , and in view of the constant Neumann data this means that has constant mean curvature. Therefore is a ball of radius .

As explained in Reference 5, for the right -function is , and annuli are cases in which the overdetermined problem has viscosity solutions of class . The case was also studied in great detail in a series of papers by Crasta and Fragalá, who relaxed the smoothness of the boundary, see e.g. Reference 7.

The normalized -Laplacian has also been studied in the context of evolution equations in a number of papers, see Reference 11Reference 8Reference 4Reference 3Reference 20Reference 12Reference 10.

2. Definitions and Comparison Result

In the notation of the theory of viscosity solutions we study the equation

Definition 2.1.

Following Reference 6, is a viscosity solution of the equation , if it is both a viscosity subsolution and a viscosity supersolution.

is a viscosity subsolution of , if for every and such that has a minimum at , the inequality holds. Here is the lower semicontinuous hull of .

is a viscosity supersolution of , if for every and such that has a maximum at , the inequality holds. Here is the upper semicontinuous hull of .

If denotes a symmetric real valued matrix, we denote its eigenvalues by . Using this notation, it is a simple exercise to find out that

so

while

that is

The following comparison principle has been derived in Reference 18Reference 17.

Proposition 2.2.

Suppose and are in and are viscosity super- resp. subsolutions of on a domain and on . Then in .

It can be used to show the positivity of and a Hopf Lemma.

Lemma 2.3.

Suppose satisfies a uniform interior sphere condition and is a viscosity solution of in such that on . Then is positive in and there exists a number such that for all

Here denotes the outward unit normal at . In fact, one can compare to a radially symmetric and radially decreasing solution on the interior of the sphere. On a ball solutions of the Dirichlet problem for are unique by Proposition 2.2, so they are necessarily radial. In polar coordinates turns into the tractable ODE Reference 15

with as boundary conditions, and this boundary value problem has the explicit solution

so that is positive in every ball with radius contained in . Moreover, Lemma 2.3 holds with .

3. Proof of Main Result

To prove Theorem 1.1 we follow an idea developed in Reference 1. We first note that from the regularity result stated in Theorem 4.2 in the Appendix, we have that is in for some and therefore the Neumann condition is realized in the classical pointwise sense. Now because by assumption on and , we have that in an -neighborhood of inside defined by . Therefore the operator is well-defined in the classical sense in . Moreover in , since , we have that solves

where

is uniformly elliptic and is in . Consequently, by the classical Schauder theory we can assert that is of class in . We now move a hyperplane, say from the left by the amount in -direction into and compare the original solution to the reflected one in the reflected cap. By the weak comparison principle, Proposition 2.2, we know that in the reflected cap . Moreover, since in , we have that solve an equation in of the form

where is uniformly elliptic and smooth in its arguments. Therefore solves the following linearized equation in ,

where

and

Moreover is uniformly elliptic and the first order coefficients are bounded in . Note that over here, we think of as a function of the matrix and the vector . Therefore, is to be thought of as the partial derivative of with respect to the coordinate in and is the partial derivative with respect to the coordinate in .

Since solves the uniformly elliptic PDE Equation 3.2 in , by the classical strong maximum principle applied to , we get that in and on the plane . The latter inequality follows from the classical Hopf Lemma applied to . We continue to move the hyperplane across. Even if the operator might become degenerate because we pass a critical point of , the weak comparison principle continues to hold, so that in the reflected cap, until one of the following cases occurs.

i) The hyperplane and meet under a right angle in a point .

ii) The reflected cap touches from the inside of in a point .

In case i) we can apply the strong maximum principle again and conclude that either in the reflected cap intersected with an neighborhood of the point , or there. But by Serrin’s corner lemma applied to which solves Equation 3.2, the first case is ruled out. In fact because the normal derivatives coincide there and the tangential ones vanish. So a partial derivative of in any direction must vanish there. Let us see what happens to second partial derivatives in direction , where is a unit vector tangent to at . We claim that

In fact in one can rewrite the differential equations for and as (see Reference 14)

where is the mean curvature of the boundary. Since , we conclude that . For the same reason, . Now , where denotes the curvature of in direction , so that . Finally one can observe that , where denotes arclength along the curve that is cut out of by the plane spanned by and . Since and are constant on , we have , and since , this completes the proof of (Equation 3.3). Therefore we can conclude that and at this point, Serrin’s corner lemma implies that in .

In case ii) we can also conclude that either in the reflected cap intersected with an neighborhood of the point or . Now in the former case, i.e., when in the reflected cap, we recall again that solves the uniformly elliptic PDE Equation 3.2 in , but then by Hopf’s Lemma applied to , we get that at , which contradicts the fact that satisfies constant Neumann data. Consequently, we have that in .

In both cases and are locally Steiner-symmetric in direction . To see that they are also globally symmetric, one can argue as follows. For reasons of continuity the set of points in which the reflected boundary coincides with the original boundary is closed in . But it is also open in . In fact in any point that belongs to the boundary of this intersection one can apply the corner lemma again to see that a whole neighborhood still belongs to it.

Since this Steiner symmetry happens in any direction, we can conclude that is a ball and is radial and radially decreasing.

4. Appendix

In this section, we state and prove a basic regularity result which has been referred to in the proof of Theorem 1.1 in the previous section. In order to do so, we first introduce the relevant notion of extremal Pucci type operators. Let and denote the maximal and minimal Pucci operators corresponding to , i.e., for every we have

where