Work

Nonlocal operators in divergence form and existence theory for integrable data

Year: 2025

Type: article

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 $L^1(Ω)$ 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 $L^1(Ω)$ 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 $s\nearrow 1$, every classical operator in divergence form.

Source: Journal of Functional Analysis

DOI: 10.1016/j.jfa.2025.111317

Authors: David Arcoya, Serena Dipierro, Edoardo Proietti Lippi, Caterina Sportelli, Enrico Valdinoci

Language: English

Topic: Nonlinear Partial Differential Equations

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,250