Lipschitz Stability and Hadamard Directional Differentiability for Elliptic and Parabolic Obstacle-Type Quasi-Variational Inequalities

TL;DR

This paper establishes Lipschitz stability and Hadamard directional differentiability for solutions of obstacle-type quasi-variational inequalities, applicable to elliptic and parabolic problems without restrictive assumptions.

Contribution

It introduces new conditions under which solution mappings are Lipschitz continuous and Hadamard directionally differentiable, extending analysis to broader classes of problems.

Findings

Maximal and minimal solutions depend Lipschitz continuously on parameters.

Directional derivatives characterized via linearized fixed-point equations.

Results apply to impulse control and boundary control problems.

Abstract

This paper is concerned with the sensitivity analysis of a class of parameterized fixed-point problems that arise in the context of obstacle-type quasi-variational inequalities. We prove that, if the operators in the considered fixed-point equation satisfy a positive superhomogeneity condition, then the maximal and minimal element of the solution set of the problem depend locally Lipschitz continuously on the involved parameters. We further show that, if certain concavity conditions hold, then the maximal solution mapping is Hadamard directionally differentiable and its directional derivatives are precisely the minimal solutions of suitably defined linearized fixed-point equations. In contrast to prior results, our analysis requires neither a Dirichlet space structure, nor restrictive assumptions on the mapping behavior and regularity of the involved operators, nor sign conditions on the directions that are considered in the directional derivatives. Our approach further covers the elliptic and parabolic setting simultaneously and also yields Hadamard directional differentiability results in situations in which the solution set of the fixed-point equation is a continuum and a characterization of directional derivatives via linearized auxiliary problems is provably impossible. To illustrate that our results can be used to study interesting problems arising in practice, we apply them to establish the Hadamard directional differentiability of the solution operator of a nonlinear elliptic quasi-variational inequality, which emerges in impulse control and in which the obstacle mapping is obtained by taking essential infima over certain parts of the underlying domain, and of the solution mapping of a parabolic quasi-variational inequality, which involves boundary controls and in which the state-to-obstacle relationship is described by a partial differential equation.