:point_right: On November 7 and 14, we covered the following:

  • The Rellich–Kondrashov theorem on the compactness of imbedding of W01,p(Ω)W_0^{1,p}(\Omega) on bounded domains, for 1p<n% <![CDATA[ 1\le p < n %]]>, into Lq(Ω)L^q(\Omega), for all 1q<p% <![CDATA[ 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 L2(Ω)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.