A variational method for second order shape derivatives

TL;DR

This paper introduces a new variational approach to compute second order shape derivatives for functionals defined on Sobolev spaces, providing a general existence and representation theorem, with applications to p-torsional rigidity.

Contribution

It presents a novel method for calculating second order shape derivatives, extending the theoretical framework for shape optimization problems.

Findings

Established a general existence theorem for second order shape derivatives.

Derived a representation formula applicable to smooth convex densities.

Applied the method to the p-torsional rigidity functional for p ≥ 2.

Abstract

We consider shape functionals obtained as minima on Sobolev spaces of classical integrals having smooth and convex densities, under mixed Dirichlet-Neumann boundary conditions. We propose a new approach for the computation of the second order shape derivative of such functionals, yielding a general existence and representation theorem. In particular, we consider the p-torsional rigidity functional for p grater than or equal to 2.