The Nonlocal Inverse Problem of Donsker and Varadhan

TL;DR

This paper extends the classical inverse problem of Donsker and Varadhan to nonlocal elliptic operators, establishing a nonlocal version that incorporates a bilinear transport term, broadening the scope of inverse problems in nonlocal PDEs.

Contribution

It introduces a nonlocal inverse problem framework for elliptic operators with a nonlocal transport term, generalizing the classical local results.

Findings

Proves a nonlocal inverse problem analogous to Donsker and Varadhan's classical result.

Establishes conditions under which the nonlocal inverse problem is well-posed.

Extends the theory of inverse problems to a broader class of nonlocal operators.

Abstract

In this paper we prove a nonlocal version of the celebrated Inverse Problem of Donsker and Varadhan~\cite{DV} for nonlocal elliptic operators of the form $$ \cL u = L_K u + \cB_K (h,u), $$ where $L_K$ is a uniformly elliptic nonlocal operator with smooth coefficients, and, for $h:\R^N \to \R$ smooth and bounded, $\cB_K(h,u)$ is the associated bilinear form, which can be regarded as a nonlocal transport term.