Existence of surfaces optimizing geometric and PDE shape functionals under reach constraint

TL;DR

This paper proves the existence of hypersurfaces that minimize shape functionals under reach constraints, extending previous results to include PDE-related costs with simpler proofs and broader applicability.

Contribution

It provides a unified, simplified framework for existence results of shape optimization under reach constraints, including new PDE-related problems.

Findings

Existence of shape minimizers under reach constraints.

Extension of results to PDE-involving cost functionals.

Simplified proofs using signed distance functions.

Abstract

This article deals with the existence of hypersurfaces minimizing general shape functionals under certain geometric constraints. We consider as admissible shapes orientable hypersurfaces satisfying a so-called reach condition, also known as the uniform ball property, which ensures C 1,1 regularity of the hypersurface. In this paper, we revisit and generalise the results of [9, 4, 5]. We provide a simpler framework and more concise proofs of some of the results contained in these references and extend them to a new class of problems involving PDEs. Indeed, by using the signed distance introduced by Delfour and Zolesio (see for instance [7]), we avoid the intensive and technical use of local maps, as was the case in the above references. Our approach, originally developed to solve an existence problem in [12], can be easily extended to costs involving different mathematical objects associated with the domain, such as solutions of elliptic equations on the hypersurface.