Nonlocal operators in divergence form and existence theory for integrable data

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[jls-end-space/], 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[jls-end-space/], every classical operator in divergence form.