Shape Optimization of an Elliptic Equation in an Exterior Domain

This work concerns the shape optimization problem governed by some elliptic equations in exterior domains. The existence of optimal solutions is obtained and the so-called $\stackrel{\wedge}{\Gamma}$-property for a family ${\cal O}$ of open subsets is established; i.e., if $\Omega_m, \Omega\in {\cal O}$ are such that $\Omega_m\ra \Omega$ in the Hausdorff metric, then for any $K\subset\subset R^N\setminus \o \Omega$, there exists $m(K)>0$ such that for all $m\geq m(K), K\subset\subset R^N\setminus \o \Omega_m$.