Sensitivity analysis of elliptic variational inequalities of the first and the second kind

Abstract

This thesis is concerned with the differential sensitivity analysis of elliptic variational inequalities of the first and the second kind in finite and infinite dimensions. We develop a general theory that provides a sharp criterion for the Hadamard directional differentiability of the solution operator to an elliptic variational inequality and introduce several tools that facilitate the sensitivity analysis in practical applications. Our analysis is accompanied by examples from mechanics and fluid dynamics that illustrate the strengths and limitations of the obtained results. We further establish strong and Bouligand stationarity conditions for optimal control problems governed by elliptic variational inequalities in a general setting that covers, e.g., the situations where the control-to-state mapping is a metric projection or a non-smooth elliptic partial differential equation.