Journal of Convex Analysis 14 (2007), No. 4, 807--822
Copyright Heldermann Verlag 2007
Structure of Shape Derivatives Around Irregular Domains and Applications
Jimmy Lamboley
Dep. de Mathématiques, Ecole Normale Supérieure de Cachan, Campus de Ker-Lann, 35170 Bruz, France
jimmy.lamboley@bretagne.ens-cachan.fr
Michel Pierre
Dép. de Mathématiques, Ecole Normale Supérieure de Cachan, Campus de Ker-Lann, 35170 Bruz, France
michel.pierre@bretagne.ens-cachan.fr
We describe the structure of shape derivatives around sets which are only assumed to be of finite perimeter in the real N-dimensional space RN. This structure allows us to define a useful notion of positivity of the shape derivative and we show it implies its continuity with respect to the uniform norm when the boundary is Lipschitz (this restriction is essentially optimal). We apply this idea to various cases including the perimeter-type functionals for convex and pseudo-convex shapes or the Dirichlet energy of an open set.
Keywords: Shape optimization, shape derivatives, sets of finite perimeter, convex sets, Dirichlet energy.
[ Fulltext-pdf (442 KB)] for subscribers only.