Variational methods for solving engineering problems

Abstract

The main object of the thesis is finding weak solutions of nonlinear boundary value problems with $p$-Laplacian. The approach is based on the fact that such solutions are related to critical points of the corresponding functional. In the thesis, two types of critical points are investigated.

The first part deals with critical points in the form of minimum, and two different approaches are introduced. One uses the fundamental theorem of the calculus of variations and a classical theory from the calculus of variations, while the other method is based on Ekeland's variational principle. Consequently, two alternative proofs of the existence and uniqueness of a weak solution of the Poisson equation with $p$-Laplacian and Dirichlet boundary conditions are presented.

The most important part of the thesis is devoted to various types of critical points, which are the main topic of the author's research. Three different approaches are proposed. First, the famous Mountain pass theorem and the original mountain pass algorithm are introduced. Then, a ray minimax theorem that can be considered an alternative to the classical Mountain pass theorem for certain types of problems is proposed. A complete proof and an algorithm based on the introduced method are also included. The last presented approach can even be considered an alternative to the whole mountain pass-based method. It is shown that the nontrivial solutions of the problem are related to critical points of a certain functional different from the energy functional, and some solutions correspond to its minimum. The crucial theorems, proofs, and two numerical algorithms based on the introduced approach are also proposed. All three approaches are then applied to prove the existence of nontrivial critical points of the given problem.

The final part of the thesis is devoted to numerical experiments. The first one focuses on the choice of the descent direction, which is a challenging part of the research, and the possible stopping criterion for the introduced algorithms. Consequently, the presented numerical methods are applied to the given boundary value problems and their behaviour and performance are analyzed and compared.

According to the author's opinion, one of the main contributions of the thesis is the ray mountain pass theorem, the proofs, the theoretical background and the numerical algorithm. Another significant result is the new method that is an alternative to the mountain pass-based approach, again including complete proofs. Also, the choice of the descent direction is an interesting topic that has never been adequately discussed before.