Sumários

Fourier Transform: part I

9 Março 2017, 11:00 • Hugo Tavares

Computation of the derivative of the fundamental solutions of the Laplacian in the sense of distributions.

Product of a smooth function with a distributions. Examples.

Definition of Fourier transform and Inverse Fourier transform in L^1.

The Schwartz space, definition of convergence and some continuous embeddings.

Distributions

7 Março 2017, 14:00 • Hugo Tavares

The theory of Distributions:

- Definition of the test function space, and convergence.

- Definition of distribution, and convergence in the sense of distributions.

- Derivative of a distribution.

- Examples.

The Heat Equation [E, 44-65]: fundamental solution of the heat operator and its distributional interpretation; solution of the initial-value problem; Duhamel principle and the classical solution of the nonhomogeneous Cauchy problem in the R^n; the energy method, uniqueness of generalised solutions and backwards uniqueness; the parabolic strong maximum principle; remark on the separation of variables and the general eigenvalue problem [S,23-38]; a note on the Fourier transform and the fundamental solution of the heat operator.

Harmonic functions and the classical Dirichlet problem

23 Fevereiro 2017, 11:00 • Hugo Tavares

Harmonic functions and the classical Dirichlet problem [GT, 13-16]: Mean value properties and the maximum/minimum principle for sub/superharmonic functions; the Harnack inequality; [E, 29-41]: local estimates for harmonic functions and the Liouville theorem; Green function and representation formula for solution of the Dirichlet problem for the Poisson equation; the explicit classical solutions in the half-space and in the Ball via Poisson integrals [GT, 23-27] brief introduction to the Perron method for the solution of the Dirichlet problem with continuous data on boundaries with regular points See also [S,141-147].

The Laplace Equation

21 Fevereiro 2017, 14:00 • Hugo Tavares

The Laplace Equation [E, 20-28+41-43]: Motivation; the three classical boundary value problems for the Poisson equation; classical solutions and generalised solutions; uniqueness via the energy method; the Dirichlet principle and variational problems; the fundamental solution of the Laplace operator in R^n and its distributional interpretation; solution of the Poisson equation in the whole space; basic properties of harmonic functions—mean value property, maximum principle and regularity; the strong maximal principle;