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 A _n is a symmetric positive definite matrix-valued function and μ _n is a nonnegative Borel measure (which can take infinite values on compact sets). Under the sole equicoerciveness of A _n 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 μ _n 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 A _n 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.