A new compact class of open sets under Hausdorff distance and shape optimization

TL;DR

This paper introduces a new class of open sets that is compact under the Hausdorff distance and demonstrates their application in solving shape optimization problems for elliptic equations.

Contribution

The paper establishes a new class of open sets with compactness properties and applies this to prove existence results in shape optimization for elliptic PDEs.

Findings

New class of open sets is compact under Hausdorff distance

Existence of solutions in shape optimization problems proven

Applications to elliptic equations demonstrated

Abstract

In this paper we obtain a new class of open sets, and we prove the class is compact under the Hausdorff distance, then we prove the existence of solutions of some shape optimization for elliptic equations.