Shape sensitivity of the Hardy constant involving the distance to a boundary submanifold

TL;DR

This paper studies how the Hardy constant changes when the domain or boundary submanifold is slightly deformed, focusing on its continuity and differentiability properties.

Contribution

It provides new insights into the sensitivity of the Hardy constant to boundary and submanifold perturbations, including cases of domain and submanifold deformation.

Findings

Hardy constant's continuity under domain perturbations

Differentiability of Hardy constant with respect to boundary changes

Analysis of Hardy constant when only the boundary submanifold is deformed

Abstract

We investigate the continuity and differentiability of the Hardy constant with respect to perturbations of the domain in the case where the problem involves the distance from a boundary submanifold. We also investigate the case where only the submanifold is deformed.