Global convergence of $W^{1,\infty}$-steepest descent for PDE constrained shape optimisation with semilinear elliptic equations in function space

TL;DR

This paper proves global convergence of a steepest descent method in function space for shape optimization problems constrained by semilinear elliptic PDEs, with additional results in two dimensions.

Contribution

It establishes the first global convergence proof for the steepest descent method in this PDE-constrained shape optimization setting.

Findings

Global convergence in Lipschitz topology for shape optimization

Conditional convergence results for shapes in two dimensions

Application to semilinear elliptic PDE constrained problems

Abstract

We prove global convergence in function space for the steepest descent method in shape optimisation with semilinear elliptic partial differential equations. Steepest descent is realized in the Lipschitz topology. In addition, we prove a conditional convergence result for the resulting shapes in two space dimensions.