Existence of minimizers for non-quasiconvex functionals arising in optimal design

This paper investigates the existence of minimizers for the so-called Kohn-Strang functional with affine boundary conditions. Such a functional, which arises in optimal shape design problems in electrostatics, is not quasi-convex, and therefore existence of minimizers is, in general, guaranteed only for its quasi-convex envelope. Such a quasi-convexification has been computed in two space dimensions in [11]. Recently, necessary and sufficient conditions on the affine boundary conditions for existence of minimizers for the Kohn-Strang functional have been derived in two space dimensions in [7]. We generalize these previous results for arbitrary space dimensions. Our method relies on the homogenization approach for relaxing optimal design problems. We also generalize our results to some variants of the Kohn-Strang functional.