Wulff shape characterizations in overdetermined anisotropic elliptic problems

Abstract

We study some overdetermined problems for possibly anisotropic degenerate elliptic PDEs, including the well-known Serrin's overdetermined problem, and we prove the corresponding Wulff shape characterizations by using some integral identities and just one pointwise inequality. Our techniques provide a somehow unified approach to this variety of problems.

Geometric aspects of p-capacitary potentials

“…They will be discussed in detail in Section 3. Dropping any attempt to be complete we observe that other results in the same spirit can be found for example in [55,8,29,30,41,42,46] and reference therein.…”

Serrin’s type overdetermined problems in convex cones

“…Lemma 6 (Pohozaev-type identity). Let Ω be a sector-like domain and assume that f satisfies (4). Let u ∈ W 1,∞ (Ω) be a solution to (5).…”

“…Let Ω ⊂ R N be a sector-like domain and assume that f satisfies (4). Let u ∈ W 1,∞ (Ω) be a solution of (5) such that (9) holds.…”

“…Theorem 1. Let f satisfy (4). Let Σ be a convex cone such that Σ \ {O} is smooth and let Ω be a sector-like domain in Σ.…”

Stress concentration for closely located inclusions in nonlinear perfect conductivity problems

“…We first recall that the Finsler p−Laplacian fulfills the maximum and comparison principles (see for instance [12,Lemma 2.3]). In the following lemma, we show that the maximum and minimum of u δ are attained at the boundary of Ω (and not on ∂D i δ , i = 1, 2).…”