# Suitable Spaces for Shape Optimization

## TL;DR

This paper explores the geometric structure of shape spaces to improve shape optimization methods, introducing new shape spaces and defining Riemannian metrics, gradients, and Hessians for enhanced optimization techniques.

## Contribution

It extends the shape space framework to include H^{1/2}-shapes and develops the associated Riemannian geometric tools for shape optimization.

## Key findings

- Defined Riemannian shape gradient and Hessian for Sobolev metrics.
- Extended shape space to H^{1/2}-shapes with a diffeological structure.
- Provided a foundation for optimization on diffeological spaces.

## Abstract

The differential-geometric structure of the manifold of smooth shapes is applied to the theory of shape optimization problems. In particular, a Riemannian shape gradient with respect to the first Sobolev metric and the Steklov-Poincar\'{e} metric are defined. Moreover, the covariant derivative associated with the first Sobolev metric is deduced in this paper. The explicit expression of the covariant derivative leads to a definition of the Riemannian shape Hessian with respect to the first Sobolev metric. In this paper, we give a brief overview of various optimization techniques based on the gradients and the Hessian. Since the space of smooth shapes limits the application of the optimization techniques, this paper extends the definition of smooth shapes to $H^{1/2}$-shapes, which arise naturally in shape optimization problems. We define a diffeological structure on the new space of $H^{1/2}$-shapes. This can be seen as a first step towards the formulation of optimization techniques on diffeological spaces.