Shape derivatives for minima of integral functionals

For $\mathrm{\Omega }$ varying among open bounded sets in ${\mathbb{R}}^n$, we consider shape functionals $J(\mathrm{\Omega })$ defined as the infimum over a Sobolev space of an integral energy of the kind ${\int }_{\mathrm{\Omega }}[f(\mathrm{\nabla }u)+g(u)]$, under Dirichlet or Neumann conditions on $\mathrm{\partial }\mathrm{\Omega }$. Under fairly weak assumptions on the integrands $f$ and $g$, we prove that, when a given domain $\mathrm{\Omega }$ is deformed into a one-parameter family of domains $\Omega _\varepsilon$throughaninitialvelocityfield$V\in W ^ {1, \infty} ({\mathbb R} ^n, {\mathbb R} ^n)$,thecorrespondingshapederivativeof$J$at$\Omega$inthedirectionof$V$ 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 $\mathrm{\partial }\mathrm{\Omega }$. 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.