Faber–Krahn Inequalities for the Robin-Laplacian: A Free Discontinuity Approach

Abstract

We prove a full range of Faber-Krahn inequalities in a nonlinear setting and for non smooth domains, including the open case of the torsional rigidity. The key point of the analysis relies on regularity issues for free discontinuity problems in spaces of functions of bounded variation. As a byproduct, we obtain the best constants for a class of Poincaré inequalities with trace terms in euclidean spaces

The Saint-Venant Inequality for the Laplace Operator with Robin Boundary Conditions

“…The complete proof of Theorem 2.1 with the full comprehension of the non-smooth setting was given in [10] and is based on a free discontinuity approach. In this section, we shall give the main ideas of the proof in the case q = 1.…”

“…This choice brings a little simplification of the exposition. We refer to the full proof and to the general case to reference [10]. Before any technical attack of the problem, let us give the principle of the proof, which can be stated as "Existence and regularity of the optimal set =⇒ it is the ball", The proof of this principle relies on reflection arguments which will be described below.…”

“…In view of (3.4), this entails that 0 ∈ Ω opt , so that Ω opt contains a ball centered at the origin. Denoting by B the maximal ball contained in Ω opt , one shows (see [10]) that if Ω opt does not coincides with B , then B is more convenient for E β : the comparison between E β (B ) and E β (Ω opt ) is easily exploited by restricting ψ(|x|) on B . The case 0 ∈ Ω opt leads with similar arguments to the conclusion that Ω opt coincides with an annulus.…”

“…The complete proof of Theorem 2.1 was given in [10] in a larger setting than just the Lipschitz sets. The method is based on a free discontinuity approach to the isoperimetric inequality via the principle "existence and regularity =⇒ the optimum is the ball" (a free discontinuity approach for the linear eigenvalue is already contained in [9], but the optimality of the ball was shown by adapting the Bossel-Daners method).…”

“…The purpose of this survey article is to give the main ideas, from an intuitive perspective, of the proof of Bossel-Daners for q = 2 and of the proof given in [10] for q = 1. In fact, the restriction to q = 1 simplifies formally only the exposition of the proof of Theorem 2.1.…”