Concentration breaking on two optimization problems

TL;DR

This paper investigates boundary concentration breaking phenomena in two thermal insulation problems on Lipschitz domains, utilizing spectral analysis, regularity results, and perturbation techniques to identify thresholds and specific breaking values.

Contribution

It introduces new analysis of concentration breaking in thermal insulation problems, including regularity results and explicit thresholds on Lipschitz and ball domains.

Findings

Identification of boundary concentration breaking thresholds.

Global Hölder regularity of minimizers on Lipschitz domains.

Explicit breaking values related to 2π in 2D.

Abstract

In the present paper, we study the boundary concentration breaking phenomena on two thermal insulation problems considered on Lipschitz domains, based on Serrin's overdetermined results, perturbation argument and comparison of Laplacian eigenvalues with different boundary conditions. Since neither of the functionals in the two problems is $C^1$, another key ingredient is to obtain the global H\"older regularity of minimizers to both problems on Lipschitz domains. Also, exact dependence on domain of breaking thresholds is also given in the first problem, and the breaking values are obtained in the second problem on ball domains, which are related to $2\pi$ in dimension $2$.