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 $L^1(\Omega )$ and to be suitably dominated. We also prove that the solution that we find converges, as $s\nearrow 1$, 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 $L^1(\Omega )$ 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 $s\nearrow 1$, every classical operator in divergence form.

References in corpus (6)

• Dirichlet forms and semilinear elliptic equations with measure data
• Integro-Differential Elliptic Equations
• On a fractional Monge-Ampère operator
• An existence theory for superposition operators of mixed order subject to jumping nonlinearities
• Renormalized solutions of semilinear equations involving measure data and operator corresponding to Dirichlet form
• Dirichlet problem for semilinear partial integro-differential equations: the method of orthogonal projection