Structure of Shape Derivatives Around Irregular Domains and Applications

We describe the structure of shape derivatives around sets which are only assumed to be of finite perimeter in the real N-dimensional space $\mathbb{R}^N$. 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.