A Convergence Criterion for Elliptic Variational Inequalities

TL;DR

This paper establishes a convergence criterion for solutions to elliptic variational inequalities with unilateral constraints, providing necessary and sufficient conditions for sequence convergence and illustrating applications in heat transfer and contact problems.

Contribution

It introduces a new convergence criterion for elliptic variational inequalities, unifying and extending existing results, with practical applications in physics and engineering.

Findings

Provides necessary and sufficient conditions for convergence.

Recovers classical convergence and well-posedness results.

Demonstrates applications in heat transfer and contact mechanics.

Abstract

We consider an elliptic variational inequality with unilateral constraints in a Hilbert space $X$ which, under appropriate assumptions on the data, has a unique solution $u$. We formulate a convergence criterion to the solution $u$, i.e., we provide necessary and sufficient conditions on a sequence $\{u_n\}\subset X$ which guarantee the convergence $u_n\to u$ in the space $X$. Then, we illustrate the use of this criterion to recover well-known convergence results and well-posedness results in the sense of Tykhonov and Levitin-Polyak. We also provide two applications of our results, in the study of a heat transfer problem and an elastic frictionless contact problem, respectively.