A Talenti Comparison Result for Solutions to Elliptic Problems with Robin Boundary Conditions

Abstract

Comparison results of Talenti type for elliptic problems with Dirichlet boundary conditions have been widely investigated in recent decades. In this paper, we deal with Robin boundary conditions. Surprisingly, contrary to the Dirichlet case, Robin boundary conditions make the comparison sensitive to the dimension, and while the planar case seems to be completely settled, in higher dimensions some open problems are yet unsolved. © 2023 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.

A Rigidity Result for the Robin Torsion Problem

“…In the recent paper [4], the authors obtain the same result using symmetrization techniques. They establish a Talenti-type comparison result between suitable Lorentz norms of the solution to the following problems:…”

“…The starting point of the proof of our main results is the following Lemma, proved in [4]. For the convenience of exposition, we report here the proof.…”

“…Remark 5 Integrating ( 13) and ( 14) between 0 and t and integrating by parts, it is proved in [4] that…”

“…On the other hand, in the case n = 2, the Open Problem 1 contained in [3] is solved in [4] in the following stronger version: where u is the Schwartz rearrangement of the solution to (1) and v is the solution to…”

“…The idea of the proof is the following. Starting from the proof in [4] of the pointwise comparison (4), we show that the equality u = v implies that the level sets of u are balls on the boundary of which the normal derivative of u is constant. Then, we prove that these balls are concentric, using an argument inspired by [11] (see also [12,Lemma 6]).…”