A nonlocal transmission problem on a hybrid continuous-discrete domain

TL;DR

This paper investigates a quadratic nonlocal variational problem on a hybrid domain combining continuous and discrete elements, establishing existence, uniqueness, and characterizing the solution through a coupled hybrid system.

Contribution

It introduces a novel hybrid variational framework and proves coercivity, existence, and uniqueness of solutions for the combined continuous-discrete nonlocal problem.

Findings

The interface term provides a coercive coupling between phases.

Existence and uniqueness of the minimizer are established.

The solution is characterized by a coupled hybrid Euler--Lagrange system.

Abstract

We study a quadratic nonlocal variational problem on a hybrid domain formed by a compact interval and finitely many discrete points. The associated energy splits into continuous, discrete, and interface contributions. Our main estimate shows that the interface term yields a coercive coupling between the two phases and provides an equivalent hybrid norm. As a consequence, we prove existence and uniqueness of a minimizer for the corresponding variational problem and characterize it as the unique weak solution of the associated hybrid Euler--Lagrange system. The latter combines a nonlocal integral equation on the continuous component with a finite nonlocal algebraic system on the discrete nodes.