A free boundary problem with multiple boundaries for a general class of anisotropic equations☆
Introduction
Let N ≥ 2, be a positive homogeneous function of degree 1, with the following properties:Obviously, since F is homogeneous and defined at the origin, we have . Furthermore, we also assume that F satisfies the following convexity condition:Next, let be a function such thatAs in Cozzi et al. [8], we assume that either conditions (I) or (II) is verified, where:
(I) There exists such that, for any we have
(II) The composition G ∘ F is of class and for any R > 0 there exists a positive constant ϱ > 0 such that, for any with |ξ| ≤ R, we have
Now, let us introduce the following quasilinear anisotropic operator:We immediately notice that Q coincides with the anisotropic p-Laplace operator in the particular case while when Q represents the so-called anisotropic mean curvature operator. Let Ωk, be m strictly convex regions in strictly contained in a bounded domain . In this paper we are concerned with the following free boundary problem:where the boundaries ∂Ωk, are assumed to be of class C2,α. We note that in problem (1.7) the values of the positive constants ck are not given, but they are determined from the further conditions:where |Ωk| is the N-volume of Ωk and ν is the inward unit normal to ∂Ωk. We also note that the assumptions on F and G considered in this paper guarantee that where (see Cozzi et al. [8], Propositions 3.1 and 3.2). Therefore, the partial derivatives of u(x), up to third order, are all well defined on .
The main result of this paper states the following: Theorem 1.1 If problem (1.7)and (1.8) has a weak solution satisfyingthen Ω0 contains a single hole Ω1 and, up to a translation, Ω0 and Ω1 are concentric Wulff shapes whose radii are given by . Moreover, the solution u(x) is given explicitly by the following formula:where is the inverse of the map t → G′(t) on (0, ∞) and .
The outline of this paper is as follows. In Section 2 we briefly give the definition of the Wulff shape of F and mention some properties of F and the anisotropic mean curvature (or F-mean curvature) of the level surfaces of u(x). More precisely, we will first define the anisotropic mean curvature of a level surface of u(x) and give a representation of the anisotropic operator Q in terms of the anisotropic mean curvature of the underlying level surface of u. Furthermore, an anisotropic version of Minkowski inequality and a Newton type inequality for symmetric matrices will be also recalled. The proof of the main result is given in Section 3. The main ingredients of the proof are as follows: a maximum principle for an appropriate functional combination of u(x) and ∇u(x), i.e. a P–function in the sense of Payne (see the book of Sperb [14]), a Rellich type identity and some properties of the given P-function and geometric arguments involving the anisotropic curvature of the free boundary. We note that this idea of proof, based on maximum principles for P-functions, was developed first by Weinberger in [15] and used recently to investigate other anisotropic problems by several authors (see Wang and Xia [16], Lv et al. [12] or Barbu and Enache [1], [2], [3]). For an account on other similar results for anisotropic free boundary problems, using different method of proof, we refer the reader to Cianchi and Salani [7], the survey paper of Farina and Valdinoci [9], respectively the recent articles of Bianchini and Ciraolo [5] and Bianchini et al. [6].
Finally, for convenience, notice that throughout this paper the comma is used to indicate differentiation and the summation from 1 to N is understood on repeated indices. Moreover, we adopt the following notations:where
Section snippets
Preliminaries
In this section we are going to recall some properties which play an important role in the proof of our result.
First, let F∘ be the dual norm of F, that isalso called the polar of F. We denote by WF(1) and the unitary balls with respect to the F, respectively F∘, that isIn general, for r > 0, we say that is the Wulff shape (or equilibrium crystal shape) of F, of radius rand center 0. A set
The proof of Theorem 1.1
Let us first consider the following P-function:where u(x) is a solution to Eq. (1.7)1. We then have the following maximum principle: Lemma 3.1 The auxiliary function P(u; x) defined in (3.1) takes its maximum value on ∂Ω, unles P ≡ const. in Ω. Proof of Theorem 1.1 The main idea of the proof is the construction of an elliptic second order differential inequality for the auxiliary function P(u; x). The conclusion of the theorem will then follow immediately, as a direct consequence of
Acknowledgements
The second author was supported by a Seed Grant from the American University of Sharjah.
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This paper is dedicated to Prof. Gérard A. Philippin, on the occasion of his 75th birthday.