On Pompeiu's-Schiffer's Conjectures from Shape Optimization

TL;DR

This paper revisits Schiffer's and Pompeiu's conjectures using shape optimization, maximum principles, and symmetry methods, proposing new conditions and approaches for potential solutions and numerical analysis.

Contribution

It introduces a Riemannian infinite-dimensional framework and new sufficient conditions to analyze these classical conjectures.

Findings

Proposes a Riemannian approach for shape analysis.

Identifies conditions under which conjectures may hold.

Suggests methods for numerical simulations of domain shapes.

Abstract

Our aim is to do a come back on Schiffer's and Pompeiu's conjectures with shape optimization tools, maximum principles and Serrin's symmetry method. We propose a way to get affirmative answers in some cases. We propose also sufficient conditions thanks to Riemannian approach of infinite dimension that could be useful for numerical simulations of the shape of domains related to these conjectures.