Week 1:Test functions Distributios, calculus of distributions, support and singular support of distributions.
Week 2:Convolutions of functions, convolution of distributions, Fundamental solutions.
Week 3:Fourier transform, Fourier inversion, tempered distributions.
Week 4:Sobolev spaces, definition, approximation by smooth functions.
Week 5:Extension theorems, Poincare inequality, Imbedding theorems.
Week 6:Compactness theorems, trace theory.
Variational problems in Hilbert spaces and Lax-Milgram lemma. Examples of weak formulations of elliptic boundary value problems.Week 8:
Regularity, Galerkin’s method, Maximum principles.
Eigenvalue problems, introduction to the finite element method.
Semigroups of operators. Examples, basic properties, Hille-Yosida theorem.
Maximl dissipative operators, regularity
Week 12:Heat equation, wave equation, Schrodinger equation. Inhomogeneous equations.