Discretization of Coefficient Control Problems with a Nonlinear Cost in the Gradient

Abstract

We consider a control problem of optimal design consisting in mixing two electric phases in order to minimize a given objective function. For simplicity, we assume that the two phases are isotropic, although the results still hold true for more general composites (see [CaEtAl08]). Mathematically the problem can be formulated as follows.

Let Ω be a bounded smooth open set in $\mathbb{R}^N, N \geq 2,$, let α, β, k be positive constants such that $\alpha < \beta, k < |\Omega|$, and let f be in L ²(Ω). We look for a measurable set ω ⊂ Ω with |ω| = k such that the solution u of

$$\begin{cases}-{\rm div} (\alpha\chi_\omega + \beta \chi_{\Omega\backslash\omega})\nabla u = f \ \ {\rm in} \ \Omega,\\ u \in H^1_0(\Omega),\end{cases}$$

minimizes the functional

$$J(u) = \int_\Omega F(\nabla u)dx + G(u),$$

where $F : \mathbb{R}^N \to \mathbb{R}^N$ has a growth of order two at infinity and $G : H^1_0(\Omega)\to \mathbb{R}$ is sequentially continuous in the weak topology of $H^1_0(\Omega).$.