Estimates of First and Second Order Shape Derivatives in Nonsmooth Multidimensional Domains and Applications

TL;DR

This paper studies the continuity of first and second order shape derivatives in nonsmooth multidimensional domains for elliptic PDEs, providing sharp regularity results and applications to shape optimization with convexity constraints.

Contribution

It introduces a novel approach to analyze shape derivatives in nonsmooth domains, extending the class of applicable functionals and PDEs, and applies these results to shape optimization problems.

Findings

Sharp continuity results for shape derivatives in nonsmooth domains

Extension of estimates to broader classes of functionals and PDEs

Applications to shape optimization under convexity constraints

Abstract

In this paper we investigate continuity properties of first and second order shape derivatives of functionals depending on second order elliptic PDE's around nonsmooth domains, essentially either Lipschitz or convex, or satisfying a uniform exterior ball condition. We prove rather sharp continuity results for these shape derivatives with respect to Sobolev norms of the boundary-traces of the displacements. With respect to previous results of this kind, the approach is quite different and is valid in any dimension $N\geq 2$. It is based on sharp regularity results for Poisson-type equations in such nonsmooth domains. We also enlarge the class of functionals and PDEs for which these estimates apply. Applications are given to qualitative properties of shape optimization problems under convexity constraints for the variable domains or their complement.