A non-local coupling model involving three fractional Laplacians

Abstract

In this article the authors study a non-local diffusion problem that involves three different fractional laplacian operators acting on two domains. Each domain has an associated operator that governs the diffusion on it, and the third operator serves as a coupling mechanism between the two of them. The model proposed is the gradient flow of a non-local energy functional. In the first part of the article we provide results about existence of solutions and the conservation of mass. The second part is devoted to study the asymptotic behaviour of the solutions of the problem when the two domains are a ball and its complementary. Fractional Sobolev inequalities in exterior domains are also provided.