On a free boundary problem for a class of anisotropic equations

Abstract

This paper deals with a certain condenser capacity in an anisotropic environment. More precisely, we are going to investigate a free boundary problem for a class of anisotropic equations on a ring domain normalΩ:=normalΩ0∖falsenormalΩ¯1⊂double-struckRN,N≥2. Our aim is to show that if the problem admits a solution in a suitable weak sense, then the underlying domain Ω is a Wulff‐shaped ring. The proof makes use of a maximum principle for an appropriate P‐function, in the sense of L. E. Payne, a Rellich type identity and some geometric arguments involving the anisotropic mean curvature of the free boundary. Copyright © 2016 John Wiley & Sons, Ltd.

On a free boundary value problem for the anisotropic N-Laplace operator on an N−dimensional ring domain

“…In our paper, the usual euclidian norm of the gradient is replaced with an arbitrary norm F, satisfying assumption (1.1). The same problem for the case of the anisotropic p -Laplace operator, 1 < p < N, has already been investigated by L. Barbu-C. Enache in [2], thus the main result of this paper (Theorem 1.1) looks somehow complementary. Studying this class of anisotropic equations could have numerous applications in physics, ranging from some well-established models of surface energies in metallurgy, crystallography, and crystalline fracture theory, to noise-removal procedures in digital image processing (see [3,4,5,6,7,8,9,20,21,22,26] and references therein).…”

“…For the proof of Lemma 2.2, we refer the reader to L. Barbu-C. Enache [2], Lemma 2.2. The proof of Lemma 2.1 is mainly based on the construction of an elliptic differential inequality for the P (u; •)−function defined in (2.1) − (2.2) (for computations of this kind see also [1] and [2]).…”

Stress concentration for closely located inclusions in nonlinear perfect conductivity problems

“…We mention that there are other maximum principles for H(∇u) available in literature. In particular, one can prove that (3.24) LH(∇u) p ≥ 0 (see for instance [20] and [11]). Since L is associated to the p-Laplace equation, (3.24) may appear to be more natural to be considered.…”