Stability and Regularization of Quasi-Variational Inequalities under Monotone Operator Perturbations

TL;DR

This paper develops a comprehensive stability framework for quasi-variational inequalities under monotone operator perturbations, including convergence analysis, regularization, and applications to complex nonlinear problems in science and engineering.

Contribution

It introduces a unified stability theory for QVIs under perturbations, extending classical monotone operator results to non-monotone and regularized settings with explicit convergence rates.

Findings

Strong convergence of solutions under operator perturbations

Explicit convergence rates for regularization and approximations

Applicability to p-Laplacian, elliptic regularizations, and control problems

Abstract

This paper establishes comprehensive stability results for quasi-variational inequalities (QVIs) under monotone perturbations of the governing operator. We prove strong convergence of both minimal and maximal solutions when sequences of operators converge pointwise while preserving fundamental properties including homogeneity, strong monotonicity, Lipschitz continuity, and T-monotonicity. Our analysis extends to regularization techniques, finite-dimensional approximations, and non-monotone nonlinearities, providing explicit convergence rates under appropriate conditions. The theory encompasses applications to $p$-Laplacian operators, elliptic regularizations, and optimal control problems with QVI constraints. By developing a unified framework that bridges classical monotone operator theory with contemporary computational challenges, this work provides essential mathematical foundations for the robust numerical approximation and sensitivity analysis of quasi-variational problems arising in materials science, physics, and engineering applications.