Compactness of sequences of two-dimensional energies with a zero-order term. Application to three-dimensional nonlocal effects
Résumé
In this paper we study the limit, in the sense of the Gamma-convergence, of sequences of two-dimensional energies of the type integral(Omega) A(n)del u center dot del u dx + integral(Omega) u(2)d mu(n), where A(n) is a symmetric positive definite matrix-valued function and mu(n) 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 (F) over cap (u) + integral(Omega) u(2)d mu, where (F) over cap F is a diffusion independent of mu(n) and mu is a nonnegative Borel measure which does depend on (F) over cap. 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 newapproach 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.