Shape Optimisation with $W^{1,\infty}$: A connection between the steepest descent and Optimal Transport

TL;DR

This paper explores the connection between shape optimisation using $W^{1, abla}$ and optimal transport, introducing a Lipschitz-based approach and demonstrating its effectiveness through numerical experiments.

Contribution

It establishes a novel link between shape optimisation and optimal transport via Lipschitz functions, highlighting differences between Lipschitz and $W^{1, abla}$ semi-norms.

Findings

Lipschitz approach provides a new perspective on shape optimisation.

Numerical experiments demonstrate the practical applicability of the method.

The connection to optimal transport offers theoretical insights into shape descent directions.

Abstract

In this work, we discuss the task of finding a direction of optimal descent for problems in Shape Optimisation and its relation to the dual problem in Optimal Transport. This link was first observed in a previous work which sought minimisers of a shape derivative over the space of Lipschitz functions which may be closely related to the $\infty$-Laplacian. We provide some results of shape optimisation using this novel Lipschitz approach, highlighting the difference between the Lipschitz and $W^{1,\infty}$ semi-norms. After this, we provide an overview of the necessary results from Optimal transport in order to make a direct link to the Shape optimisation of star-shaped domains. Demonstrative numerical experiments are provided.