*Published Paper*

**Inserted:** 10 aug 2018

**Last Updated:** 10 aug 2018

**Journal:** Math. Program., Ser. B

**Volume:** 148

**Pages:** 111–142

**Year:** 2014

**Doi:** 10.1007/s10107-013-0712-6

**Abstract:**

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

**Download:**