# Partial Differential Equations

## Objectives

Students must become acquainted with the more usual methods

in linear partial differential equations and be aware of its main applications.

## General characterization

11590

6.0

##### Responsible teacher

Magda Stela de Jesus Rebelo

##### Hours

Weekly - 4

Total - Available soon

Português

### Prerequisites

Background in Functional Analysis and Measure Theory.

### Bibliography

1. L.C. Evans, Partial Differential Equations. American Mathematical Society.
2. H. Brezis, Analyse Fonctionnelle: théorie et applications. Masson.
3. D. Gilbarg & N. Trudinger, Elliptic Partial Differential Equations of Second Order. Springer.
4. J.-L. Lions & E. Magenes, Nonhomogeneous boundary value problems and Applications, vol.I a III. Springer

Available soon

Available soon

## Subject matter

1.      The classical Linear Partial Differential Equations

1.1.   Laplace’s equation

1.1.1.Fundamental solution: Derivation of the fundamental solution; Poisson’s equation; Mean-Value formulas.

1.1.2.Properties of harmonic functions: Strong maximum principle; Uniqueness; Regularity; Liouville’s theorem; Analyticity; Harnack’s inequality.

1.1.3.Green’s function: Green’s function for a half-space; Green’s function for a ball.

1.1.4.Energy methods: Dirichlet’s principle.

1.2.   Heat Equation

1.2.1.Fundamental solution: Derivation of the fundamental solution; Initial-Value problem; Nonhomogeneous problem; Mean-value formula

1.2.2.Properties of solutions: Strong maximum principle, Uniqueness;  Regularity; Local estimates

1.2.3.Energy methods

1.3.   Wave equation

1.3.1.Solution by spherical means: Alembert’s formula; Kirchhoff’s and Poisson’s formulas

1.3.2.Nonhomogeneous problem

1.3.3.Energy methods

2.      Sobolev Spaces

2.1.   Sobolev Spaces

2.1.1.Weak derivatives

2.1.2.Definition of Sobolev spaces

2.1.3.Elementary properties

2.2.   Approximation by regular functions

2.3.   Extension Theorem

2.4.   Trace Theorem

2.5.   Sobolev inequalities

2.6.   Compactness

2.7.   Poincaré’s inequalities

2.8.   Fourier Transform methods

3.      Second-Order Elliptic Equations

3.1.   Definitions

3.2.   Existence of weak solutions

3.2.1.Lax-Milgram Theorem

3.2.2.Energy estimates

3.2.3.Fredholm alternative

3.3.   Regularity

3.4.   Maximum principles

3.4.1.Weak maximum principle

3.4.2.Strong maximum principle

3.4.3.Harnack’s inequality

3.5.   Eigenvalues and eigenfunctions

4.      Linear Evolution equations

4.1.   Second-order parabolic equations

4.1.1.Definitions

4.1.2.Existence of weak solutions

4.1.3.Regularity

4.1.4.Maximum principles

4.2.   Second-order hyperbolic equations

4.2.1.Definitions

4.2.2.Existence of weak solutions

4.2.3.Regularity

4.2.4.Propagation of disturbances

4.2.5.Semigroup theory

## Programs

Programs where the course is taught: