Study of some problems for partial differential equations

Abstract

In this thesis, we present a study of the p-Laplacian equation, which constitutes a nonlinear generalization of the classical Laplace equation. The weak formulation of the problem, subject to homogeneous Dirichlet boundary conditions, is presented. The existence and uniqueness of the weak solution are then demonstrated through two distinct approaches: the minimization of the energy functional and the application of the generalized Lax-Milgram theorem. Furthermore, an optimal control problem governed by an ill-posed p-Laplacian equation is also studied, a case in which the state equation may admit multiple solutions for a single control. For this reason, penalization and approximation problems are introduced, the necessary optimality conditions are established, and the convergence of the approximate solutions to an optimal solution of the initial problem is proven. This work provides a rigorous theoretical framework for the analysis of optimal control problems applied to non-linear systems.