The Dirichlet problem for nonlocal operators

TL;DR

This paper establishes the well-posedness of elliptic and parabolic Dirichlet problems for linear nonlocal operators, extending classical boundary value problem frameworks to nonlocal contexts with boundary data on complements of sets.

Contribution

It formulates and proves unique solvability of nonlocal Dirichlet problems within Hilbert space frameworks, adapting classical methods to nonlocal operators.

Findings

Unique solvability of nonlocal Dirichlet problems

Extension of classical boundary problem techniques to nonlocal operators

Framework applicable to elliptic and parabolic cases

Abstract

In this note we set up the elliptic and the parabolic Dirichlet problem for linear nonlocal operators. As opposed to the classical case of second order differential operators, here the "boundary data" are prescribed on the complement of a given bounded set. We formulate the problem in the classical framework of Hilbert spaces and prove unique solvability using standard techniques like the Fredholm alternative.