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

This article deals with the existence of hypersurfaces minimizing general shape functionals undercertain geometric constraints. We consider as admissible shapes orientable hypersurfaces satis-fying 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 Guo et al and,J. Dalphin. We provide a simpler framework and more concise proofs of some of the results con-tained in these references and extend them to a new class of problems involving PDEs. Indeed, byusing the signed distance introduced by Delfour and Zolesio, we avoid the intensive and technicaluse of local maps, as was the case in the above references. Our approach, originally developedto solve an existence problem in a recent work by the same authors dedicated to optimal shapeissues for Plasma Physics, can be easily extended to costs involving different mathematical objectsassociated with the domain, such as solutions of elliptic equations on the hypersurface