Shape derivatives for minima of integral functionals

For Ω varying among open bounded sets in Rn, we consider shape functionals J(Ω) defined as the infimum over a Sobolev space of an integral energy of the kind ∫[f(∇u)+g(u)], under Dirichlet or Neumann conditions on ∂Ω. Under fairly weak assumptions on the integrands f and g, we prove that, when a given domain Ω is deformed into a one-parameter family of domains Ωε through an initial velocity field V∈W1,∞(Rn,Rn), the corresponding shape derivative of J at Ω in the direction of V exists. Under some further regularity assumptions, we show that the shape derivative can be represented as a boundary integral depending linearly on the normal component of V on ∂Ω. Our approach to obtain the shape derivative is new, and it is based on the joint use of Convex Analysis and Gamma-convergence techniques. It allows to deduce, as a companion result, optimality conditions in the form of conservation laws.