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

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 ${\mathcal{C}}^{1,1}$ regularity of the hypersurface. In this paper, we revisit and generalize the results of Dalphin (2018 and 2020) and Guo and Yang (2013). 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, 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 Privat, Robin, and Sigalotti’s 2022 paper, can be easily extended to costs involving different mathematical objects associated with the domain, such as solutions of elliptic equations on the hypersurface.