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.
Week 7 : 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.
Week 9 : Eigenvalue problems, introduction to the finite element method
.
Week 10 : Semigroups of operators. Examples, basic properties, Hille-Yosida theorem.
Week 11 : Maximl dissipative operators, regularity
Week 12 : Heat equation, wave equation, Schrodinger equation. Inhomogeneous equations.
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