Equações Diferenciais Parciais

Objectives

Introduction to the study of Partial Differential Equations, using classical results and modern techniques of Functional Analysis.

Syllabus

1. PDEs of Mathematical Physics. Examples. Variational Problems. Cauchy- Kowalevski theorem. 2. Introduction to elliptic equations. Laplace's equation. Maximum principle. Dirichlet problem. Mean value theorems. Poisson kernel. Harnack inequality. Perron's method. Poisson equation. Weak solutions. L2 theory for the Dirichlet problem. Generalized Dirichlet problem. Lax- Milgram theorem, Gärding inequality, compactness. Weak solutions and regularity. Maximum principle for second order equations. 3. Introduction to evolution equations. First order equations and systems of partial differential equations. Characteristics. Local existence and uniqueness. Hyperbolic second order linear equations and reduction to first-order system. Wave equation. Energy estimates and unicity. Spherical means method. Heat equation. Maximum principle. Gauss kernel. Maximum principle for second order equations.

Prerequisites

Differential and Integral Calculus III.

Cross Competence Component

The UC allows the development of transversal competences on Critical Thinking, Creativity and Problem Solving Strategies, in class, in autonomous work and in the several evaluation components. The percentage of the final grade associated with these competences should be around 15%.

Ethical Principles

All members of a group are responsible for the group’s work. In any assessment, every student shall honestly disclose any help received and sources used. In an oral assessment, every student shall be able to present and answer questions about the entire assignment and solution.