Shape derivatives for minima of integral functionals

For $$\Omega $$Ω varying among open bounded sets in $$\mathbb R ^n$$Rn, we consider shape functionals $$J (\Omega )$$J(Ω) defined as the infimum over a Sobolev space of an integral energy of the kind $$\int _\Omega [ f (\nabla u) + g (u) ]$$∫Ω[f(∇u)+g(u)], under Dirichlet or Neumann conditions on $$\partial \Omega $$∂Ω. Under fairly weak assumptions on the integrands $$f$$f and $$g$$g, we prove that, when a given domain $$\Omega $$Ω is deformed into a one-parameter family of domains $$\Omega _\varepsilon $$Ωε through an initial velocity field $$V\in W ^ {1, \infty } (\mathbb R ^n, \mathbb R ^n)$$V∈W1,∞(Rn,Rn), the corresponding shape derivative of $$J$$J at $$\Omega $$Ω in the direction of $$V$$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 $$V$$V on $$\partial \Omega $$∂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.