Optimal control of mixed local-nonlocal parabolic PDE with singular boundary-exterior data

TL;DR

This paper studies the well-posedness, regularity, and optimal control of mixed local-nonlocal parabolic PDEs with singular boundary data, providing new unified results for such complex equations.

Contribution

It introduces the first comprehensive analysis of mixed local-nonlocal PDEs with singular data, including well-posedness, regularity, and optimal control characterizations.

Findings

Established well-posedness and regularity for elliptic and parabolic problems with singular data.

Proved existence of optimal controls and derived optimality conditions.

Unified treatment of local and nonlocal PDEs with boundary-exterior conditions.

Abstract

We consider parabolic equations on bounded smooth open sets $\Om\subset \R^N$ ($N\ge 1$) with mixed Dirichlet type boundary-exterior conditions associated with the elliptic operator $\mathscr{L} \coloneqq - \Delta + (-\Delta)^{s}$ ($0<s<1$). Firstly, we prove several well-posedness and regularity results of the associated elliptic and parabolic problems with smooth, and then with singular boundary-exterior data. Secondly, we show the existence of optimal solutions of associated optimal control problems, and we characterize the optimality conditions. This is the first time that such topics have been presented and studied in a unified fashion for mixed local-nonlocal PDEs with singular data.