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

created by lamboley on 06 Jun 2016

[BibTeX]

Published Paper

Inserted: 6 jun 2016
Last Updated: 6 jun 2016

Journal: Journal of Functional Analysis
Volume: 270
Number: 7
Pages: 2616-2652
Year: 2016

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.