Quantitative stability estimates for a two-phase Serrin-type overdetermined problem

TL;DR

This paper establishes quantitative stability estimates for a two-phase Serrin-type overdetermined problem, linking it to one-phase stability and proving non-existence results for the inner problem.

Contribution

It provides the first quantitative stability estimates for two-phase overdetermined problems and connects these to one-phase stability results.

Findings

Quantitative stability estimates for two-phase problems.

Reduction of two-phase stability to one-phase stability.

Non-existence results for the inner problem.

Abstract

In this paper, we deal with an overdetermined problem of Serrin-type with respect to a two-phase elliptic operator in divergence form with piecewise constant coefficients. In particular, we consider the case where the two-phase overdetermined problem is close to the one-phase setting. First, we show quantitative stability estimates for the two-phase problem via a one-phase stability result. Furthermore, we prove non-existence for the corresponding inner problem by the aforementioned two-phase stability result.