Lectures 5 and 6

On November 7 and 14, we covered the following:

• The Rellich–Kondrashov theorem on the compactness of imbedding of $W_0^{1,p}(\Omega)$ on bounded domains, for $1\le p < n$, into $L^q(\Omega)$, for all $1\le q < p^\ast$;

• The Ritz variational method;

• Definition of a weak solution of an elliptic equation (statement, motivation etc.);

• The Lax-Milgram theorem;

• Energy estimates and existence of weak solutions;

• Examples, including weak solutions of the Poisson equation with zero Dirichlet boundary conditions;

• Compactness of the inverse of the Dirichlet laplacian on $L^2(\Omega)$ and the corollaries of the spectral theorem for the eigenvalues of the laplacian.

You will find most of this material in Evans (Chapters 5 and 6); for the Ritz variational method and a discussion of the spectral theorem applied to the eigenvalues of the laplacian you can look up my lecture notes in Polish, Krótkie wprowadzenie do równań różniczkowych cząstkowych.