Free boundary problems

Abstract

This thesis deals with the one-phase Bernoulli problem, focusing on the existence and regularity of its solutions. After establishing the necessary preliminary theory on function spaces and convergence in the first chapter, we introduce the one-phase Bernoulli problem in the second chapter, reformulating it as a minimization problem. Then, in the third chapter, we present two illuminating examples of solutions to the problem, which imply that the Lipschitz regularity is optimal. The fourth chapter proves the existence of solutions, employing the direct method of calculus of variations. Finally, the fifth chapter reveals the Lipschitz property of generalized solutions. 1