ISBN 978-3-8439-4928-6

978-3-8439-4928-6, Reihe Mathematik

Anne-Therese Rauls
Generalized Derivatives for Solution Operators of Variational Inequalities of Obstacle Type

221 Seiten, Dissertation Technische Universität Darmstadt (2021), Softcover, B5

Zusammenfassung / Abstract

This thesis characterizes generalized derivatives for solution operators of obstacle problems. We consider the mathematical formulation of these problems as variational inequalities. It is well known that for every given force term these variational inequalities possess unique solutions and that the corresponding solutions depend Lipschitz continuously on the force terms. Due to the obstacle constraint, the solution operators are, in general, nonsmooth. However, a generalization of Rademacher's theorem to sufficiently regular infinite dimensional spaces states that the Lipschitz continuous solution operators of the variational inequalities are Gâteaux differentiable on a dense subset of their domains. For such operators, generalized derivatives can serve as a substitute for the classical Gâteaux derivatives in all points of the domain. These sets of generalized derivatives at a fixed point of the domain are defined as limits of Gâteaux derivatives at approximating sequences of the domain. For the convergence of the sequence in the domain and also for the Gâteaux derivatives, different combinations of topologies can be used in infinite dimensional spaces.

The base for our analysis is the characterization of the directional derivative as the solution operator of a variational inequality established by Mignot. This enables us to obtain a representation of the Gâteaux derivatives as solution operators of variational equations or Dirichlet problems on certain quasi-open domains. These quasi-open domains depend on the contact set and the contact forces acting between the obstacle and the solution. Subsequently, the convergence of these Gâteaux derivative operators is examined for different types of obstacle problems. We also study the numerical computation of subgradients for optimal control problems with respect to obstacle problems. An error estimate is presented that can be used for the implementation of an inexact Bundle method.