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

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.