Variational Obstacle Problems: Existence, Regularity and Applications

Abstract

Obstacle problems are a class of variational problems, which consist of minimizing integral functionals over a certain set of admissible functions. Specific to obstacle problems is the constraint that the admissible functions need to lie above a given obstacle function. We establish uniqueness and existence of minimizers in Sobolev spaces for a general variety of obstacle problems. This is done by using the direct method in the calculus of variations, which relies on coercivity and weak lower semicontinuity of the functional which is aimed to be minimized. For the classical obstacle problem, which corresponds to the Dirichlet functional, we prove the optimal regularity result that states that the minimizer is locally C^{1,1} if the obstacle is C^{1,1}. This is the central theoretical result in the thesis. Lastly, we discuss applications of the classical obstacle problem to linearized elasticity, fluid flow in porous media and financial mathematics. These applications are accompanied by numerical simulations of the classical obstacle problem.