Nonlocal operators in divergence form and existence theory for integrable data

Abstract

We present an existence and uniqueness result for weak solutions of Dirichlet boundary value problems governed by a nonlocal operator in divergence form and in the presence of a datum which is assumed to belong only to L1(Ω) and to be suitably dominated. We also prove that the solution that we find converges, as s ↗1, to a solution of the local counterpart problem, recovering the classical result as a limit case. This requires some nontrivial customized uniform estimates and representation formulas, given that the datum is only in L1(Ω) and therefore the usual regularity theory cannot be leveraged to our benefit in this framework. The limit process uses a nonlocal operator, obtained as an affine transformation of a homogeneous kernel, which recovers, in the limit as s ↗ 1, every classical operator in divergence form.