Overdetermined boundary value problems with strongly nonlinear elliptic PDE ∗

We consider the strongly nonlinear elliptic Dirichlet problem in a connected bounded domain, overdetermined with the constant Neumann condition F(r u) = c on the boundary. Here F is convex and positively homogeneous of degree 1, and its polar Frepresents the anisotropic norm on R n . We prove that, if this overdetermined


Introduction and statement of the main result
Throughout this paper let F : R n → [0, +∞) be a convex function of class C 4 (R n \ {0}) which is even and positively homogeneous of degree 1, i.e., F (tξ) = |t|F (ξ), ∀ t ∈ R, ξ ∈ R n , and F ξ i = ∂F ∂ξ i . A typical example is F (ξ) = ( n i=1 |ξ i | q ) 1 q for q ∈ (1, +∞). Set W F (r) := {x ∈ R n | F * (x) = r}, where r ∈ R + and , The set W F (r) is usually said to be the Wulff shape of F . One can easily see that if F (ξ) = |ξ|, then the corresponding Wulff shape W F (r) is the standard sphere in R n . In many problems, the Wulff shape plays a role similarly to that of standard sphere.
Roughly speaking, the Wulff shape of F is the ′′ sphere ′′ associated with the norm of F * in R n , and r ∈ R + is the ′′ radius ′′ of the Wulff shape. Further details about Wulff shape can be found in [15,19,24,33] and the references therein.
In this paper, we are interested in the strongly nonlinear elliptic operators where 1 < p < +∞. It is easy to see that some special cases of (1.1) have been extensively discussed and well known to us. For F (ξ) = |ξ| and q = 2, Q is the p-Laplace operator. For F (ξ) = ( n i=1 |ξ i | p ) 1 p , Q is the pseudo-p-Laplace operator that was studied in [5]. The anisotropic elliptic operator, which was studied in [3,11,18,36], is just the operator Q when p = 2.
In [4], M. Belloni, V. Ferone and B. Kawohl obtained the symmetry of positive solutions to the problem (P.1)-(P.2) as follows. Theorem [4] If Ω ⊂ R n has the Wulff shape of F and p = n, the level sets of any solution to (P.1)-(P.2) have the Wullf shape of F .
In general, we know that the overdetermined problem (P) has no solution. From the Theorem [4], for p = n, it is not difficult to verify that problem (P) admits a solution if Ω has the Wulff shape of F .
A natural question is that, for general 1 < p < +∞, whether the following statement holds true: if (P ) admits a solution, then Ω has the Wulff shape. (1.2) Indeed, the problem of proving (1.2) is the main purpose of the overdetermined boundary value problem, which is an interesting problem while many authors used different methods to obtain a huge amount of literature. For F (ξ) = |ξ| and p = 2, the pioneering work [31] by Serrin proved that if problem (P ) admits a solution u ∈ C 2 (Ω) then necessarily the domain Ω is a ball by the moving planes method which is now well known to us. Following the method of Serrin, there are many works discussed vary types of overdetermined problems, e.g., [25,30] for the exterior domain, [2,23] for the ring-shape domain and also [28,35] for lowly regularity assumption of ∂Ω. For the same problem in [31], a totally different method to obtain the same result was discovered by Weinberger [37] whose proof is the first successful attempt to use an associated P-function. In the spirit of Weinberger [37], using the P-function, integral identity and Alexandrov theorem (see [1] or [24]), lots of results of the overdetermined problems have obtained, we refer to [17,[20][21][22]36]. There are also other alternative methods to discuss the overdetermined problems, for example, [6][7][8][9][10][11]38].
The aim of this paper is to prove (1.2) is true for general F and 1 < p < +∞. Recently, for p = 2, G. Wang and C. Xia successfully proved (1.2) in [36]. Motivated by [36], we can also prove (1.2) for general 1 < p < +∞. The method in this paper is similar to [17] and [36], where the constant mean curvature of any level set of u is obtain by using the Pohozaev identity, the maximum principle on a suitable P-function and the relationship between the operator Q and the mean curvature of any level set of u.
Before we present our main result, let us give out the definition of the weak solution. A measurable function u is called a weak solution to problem (P ) if u ∈ W 1,p 0 (Ω) and together with the condition F (∇u) = c on ∂Ω. It was observed in [4] or [34] that for any weak solution of (1.3), u ∈ C 1,α (Ω) for some 0 < α < 1. Hence the condition F (∇u) = c on ∂Ω is well defined.
The main result of this paper is as follows. Theorem 1.1 Let F : R n → [0, +∞) be a convex function of class C 4 (R n \ {0}), which is even and positively homogeneous of degree 1. Assume further that F (ξ) > 0 for any ξ = 0, and Hess(F p ) is positive definite in R n \ {0} for 1 < p < +∞. If the overdetermined boundary value problem (P ) has a weak solution in a connected bounded domain Ω ⊂ R n with ∂Ω ∈ C 4 . Then up to translation and scaling, ∂Ω is of Wulff shape.
Remark 1.1 As F (ξ) = |ξ|, Theorem 1.1 is the same as that if we take A(t) = t p−2 in [17]. Roughly speaking, Theorem 1.1 is the anisotropic version of partially results of [17]. Furthermore, if p = 2, then Theorem 1.1 is just the Theorem 1 of [36]. Remark 1.2 From the assumption in Theorem 1.1, it is easy to see that F p is strictly convex in R n , which will be used to prove the Pohozaev identity in Lemma 3.2. Remark 1.3 Since Hess(F p ) is positive definite in R n \ {0}, one can deduce that Q is a uniformly elliptic operator in any compact subsets of Ω\C, where C = {x ∈ Ω | ∇u = 0}. By virtue of F ∈ C 4 (R n \ {0}), we have by the classical elliptic regularity theory that the weak solution u in fact belongs to C 3 (Ω \ C), which implies that u ijk is well defined in Ω \ C (see Lemma 3.1). The outline of the paper is as follows. In Section 2, we give some preliminaries. In Section 3, the key result about the P-function which plays an important role to prove Theorem 1.1 is obtained. In Section 4, the main result of this paper is proved. Section 5 is our acknowledgements.
In what follows, we recall some useful results of the function F introduced in Section 1, the relationship between the Wulff shape of F and the F-mean curvature , the relationship between the operator Q and the F-mean curvature.
Firstly, we give some properties of the 1-homogeneous function F . Proposition 2.1. (see [36]) Let F : R n → [0, +∞) be a 1-homogeneous function, then the following holds: . . . , n; (iii) F * (F ξ (ξ)) = 1 and F (F * ξ (ξ)) = 1, for any ξ = 0. We denote by H F the F-mean curvature or anisotropic mean curvature. Further details about H F , we refer to [12-14, 26, 36, 39]. Now we give a result concerning the the relationship between the Wulff shape of F and H F , which shows that the Wulff shape can be characterized as a compact connected hypersurface with constant F-mean curvature. Proposition 2.2 (see [24]) Let X : M → R n be an embedded compact hypersurface without boundary in the Euclidean space. If H F (M) is constant, then up to translations and scaling, M is of Wulff shape. Remark 2.1 Proposition 2.2 is the anisotropic version of Alexandrov theorem in [1].
Let Ω be a domain in R n and u ∈ C 2 (Ω \ {x ∈ R n | ∇u = 0}). We denote by S t a level set of u, that is and assume that S t is smooth. For simplicity of notation, in the following we use F ξ i (∇u) = F i , F ξ i ξ j (∇u) = F ij , and the Roman indices follow the summation convention.
For the relationship between the operator Q and the F-mean curvature of a level set, we have Lemma 2.1 Let H F (S t ) be the F-mean curvature of the level set S t . We have Proof. The proof is similar to that of Theorem 3 in [36], the detail proof is omitted.
To investigate the overdetermined boundary value problem (P ), we start in a natural way to study the symmetry of solution to problem (P ) when ∂Ω is the Wulff shape of F .
EJQTDE, 2012 No. 10, p. 4 Then u satisfies problem (P ) and c = 1 n 1/(p−1) , the level sets S t (u) of u is the Wulff Proof. The solution to problem (P.1)−(P.2) can be found by minimizing the functional Since F p is strictly convex, the minimizer u of the functional J on W 1,p 0 (Ω) is unique. We denote by u ♯ the convex symmetrization of u, by the Pólya-Szegö inequality (see [3]), we have Notice that the minimizer u is unique, then the following holds, So we need only consider functions of the form where r = F * (x). In view of Proposition 2.1 (iii) and (2.4), we have The corresponding Euler equation of the one-dimensional problem (2.5) is We immediately have Next, we only need to check that F (∇u) = c on ∂Ω. Indeed, since Proposition 2.1 (iii) and F is positively homogeneous of degree 1, we have for any x ∈ ∂Ω. Furthermore, it is easy to find that for x ∈ S t (u), , then the level sets S t (u) of u is the Wulff shape of F . This completes the proof.

P-function
In this section, we consider the P-function defined by The motivation for studying the P-function defined by (3.1) comes from an investigation of the analogous one dimensional problem Qu = −1 in Ω ⊂ R, that is After integration of (3.4), we have Following the arguments about P-function presented in [17,20,27,32,36], we can obtain the following result.
Theorem 3.1 Let u be a weak solution to problem (P ). Then we have In order to complete the proof of Theorem 3.1, we need to prove two importance Lemmas: a maximum principle for P-function (see Lemma 3.1 below) and a Pohozaevtype integral identity of P-function (see Lemma 3.2 below). Lemma 3.1 Let u be a weak solution to the overdetermined boundary value problem (P ). Then P (x) attains its maximum on ∂Ω. Moreover, if P (x) is not constant in Ω and P (x) attains its maximum in a pointx ∈ Ω, then necessarily ∇u(x) = 0.
Proof . Notice that P ∈ C 3 (Ω \ C) (see Remark 1.3), where C = {x ∈ Ω | ∇u = 0}, the following calculations are all taken in Ω \ C. Let By some long but straightforward computations, one may obtain an elliptic inequality of second order where P i = ∂P/∂x i and L i P i denote the terms with P i (see (3.21)-(3.26) below). Rewriting (3.7) as in Ω \ C. (3.10) Since F > 0 in Ω \ C, by (3.9) and (3.10), we can deduce from (3.8) that a ij P ij +L i P i ≥ 0, in Ω \ C. Notice that 1 2 F 2 is 2-homogeneous, we have is also 0-homogeneous, i.e., we can viewā ij as a function on the compact set S n−1 . Since Hess( 1 p F p ) = (a ij ) is positive define in R n \ {0} and λ(a ij ) = F p−2 λ(ā ij ), we know that (ā ij ) is also positive define in R n \ {0}. By the 0-homogeneous and positive define of (ā ij ), it must have a uniformly positive lower(upper) bounds for all its eigenvalues. Then, there existλ,Λ > 0 such that λ|ζ| 2 ≤ā ij (ξ)ζ i ζ j ≤Λ|ζ| 2 , for any ξ = 0, ζ ∈ R n . EJQTDE, 2012 No. 10, p. 7 Using a standard convolution argument, we can find a family of Since u ∈ C 1,α (Ω), we can choose a sequence {u ǫ } in C ∞ (Ω) such that Hence,ḡ ∈ C loc (Ω \ C) for any p ≥ 1 andḡ ≥ 0 in Ω. It is easy to see that there exist vector-valued functions {L ǫ } ⊂ C 0 (Ω, R n ) and {ḡ ǫ } ⊂ C ∞ (Ω) such that L ǫ →L uniformly in any compact sets in Ω \ C, (3.15) g ǫ →ḡ in C loc (Ω \ C). (3.16) Consider now the solution P ǫ to By the uniformly ellipticity (3.13) and regularity theory, the above approximate problem admits a solution P ǫ ∈ C ∞ (Ω). In view of this, the maximum principle implies that P ǫ attains its maximum on ∂Ω, i.e., where U is any neighborhood of C.
On the other hand, the L p regularity theory shows that P ǫ is uniformly bounded in W 2,p loc (Ω \ C). Hence, by the convergence in (3.12), (3.14)-(3.16) and the compact embedding theory, there exists a subsequence of {P ǫ } such that Suppose that there exits some point x 0 ∈ Ω such that maxΩP (x) = P (x 0 ) > max ∂Ω P (x), by (3.18) we know that x 0 must belong to the interior of C. However, the interior of C is empty which follows directly from equation Qu = −1. Thus we prove (3.19). Moreover, if P (x) is not constant in Ω and P (x) attains its maximum in a pointx ∈ Ω then necessarily ∇u(x) = 0. This completes the proof of Lemma 3.1. The remaining part of this proof is need only to prove the elliptic inequality (3.7). We first calculate the derivatives up to the second order of P , (3.21) It follows form Proposition 2.1 (i) and (3.20) that and (3.23) By (3.6), Qu = −1 can be written as From (3.23) and (3.24), we have By successive differentiation of (3.24) with respect to x k , we obtain for any i. Taking derivative of (3.27) with respect to x i and summing, we obtain From (3.21), (3.24) and (3.26), we deduce that (3.29) Denote Then, (3.29) is rewritten as The term I 2 can be computed as follows, where we used (3.23).
The term I 1 can be computed as follows, then (3.35) yields where L i P i denote the terms of P i in (3.35). Now since (a ij ) is a symmetric and positive define matrix in Ω \ C, for a fixed point x, we can choose coordinates around x such that with λ i > 0 for any i. Thus (3.24) is rewritten as (3.37) From (3.36) and (3.37), we obtain which proves that (3.7) holds.
Lemma 3.2 Let u be a weak solution to the overdetermined boundary value problem (P ). Then P (x) satisfies the following identity where |Ω| is the n-dimensional volume of Ω.
Let ν F := F ξ (ν) = F ξ (∇u) on U, we deduce that ∂u ∂ν F = ∇uF ξ (∇u) = F (∇u) = c p − p n(p − 1) u 1/p := g(u) (4.1) and On one hand, (4.1) yields that and on the other hand, we have From (4.2)-(4.4), we obtain F i (∇u)F j (∇u)u ij = g(u)g ′ (u). (4.5) We denote by H F (S t ) the F-mean curvature of the level set S t , t ∈ (0, T ), T = maxΩu. So (4.1), (4.5) and Lemma 2.1 lead to Qu − (p − 1)F p−2 (∇u) ∂ 2 u ∂ν 2 F = 1 g p−1 (u) −1 − (p − 1)g p−2 (u)g(u)g ′ (u) := h(u). (4.6) The above equality just shows that every level set of u at height t between zero and T is a hypersurface of constant F-mean curvature. By Propositon 2.2, each connected component of it must be of Wulff shape, up to translations. By the same argument in [17] or [36], we may prove that S t is simply connected for any t ∈ (0, T ). Indeed, if Γť andΓť are two connected components of a particular level set Sť (ť ∈ (0, T )), then one of them must be enclosed in another and both them are of Wulff shape with the same ′′ radius ′′ . That is, Sť contain two nested Wulff shapes of equal ′′ radius ′′ , a contradiction. Hence, Sť has only one component, i.e., is simply connected.
Therefore, S t is of Wulff shape for any t ∈ (0, T ), and ∂Ω = S 0 is also of Wulff shape. This completes the proof of Theorem 1.1.

Acknowledgements
We would like to thank the referees for their careful reading of the original manuscript.