Uniform convergence of sequences of solutions of two-dimensional linear elliptic equations with unbounded coefficients
Résumé
This paper deals with the behavior of two-dimensional linear elliptic equations with unbounded (and possibly infinite) coefficients. We prove the uniform convergence of the solutions by truncating the coefficients and using a pointwise estimate of the solutions combined with a two-dimensional capacitary estimate. We give two applications of this result: the continuity of the solutions of two-dimensional linear elliptic equations by a constructive approach. and the density of the continuous functions in the domain of the Gamma-limit of equicoercive diffusion energies in dimension two. We also build two counter-examples which show that the previous results cannot be extended to dimension three. (C) 2008 Elsevier Inc. All rights reserved.