Compactness of sequences of two-dimensional energies with a zero-order term. Application to three-dimensional nonlocal effects

Abstract

In this paper we study the limit, in the sense of the Γ-convergence, of sequences of two-dimensional energies of the type \({\int_\Omega A_n\nabla u\cdot\nabla u\,dx+\int_\Omega u^2d\mu_n}\), where An is a symmetric positive definite matrix-valued function and μn is a nonnegative Borel measure (which can take infinite values on compact sets). Under the sole equicoerciveness of An we prove that the limit energy belongs to the same class, i.e. its reads as \({\hat F(u)+\int_\Omega u^2d\mu}\), where \({\hat F}\) is a diffusion independent of μn and μ is a nonnegative Borel measure which does depend on \({\hat F}\) . This compactness result extends in dimension two the ones of [11,23] in which An is assumed to be uniformly bounded. It is also based on the compactness result of [7] obtained for sequences of two-dimensional diffusions (without zero-order term). Our result does not hold in dimension three or greater, since nonlocal effects may appear. However, restricting ourselves to three-dimensional diffusions with matrix-valued functions only depending on two coordinates, the previous two-dimensional result provides a new approach of the nonlocal effects. So, in the periodic case we obtain an explicit formula for the limit energy specifying the kernel of the nonlocal term.