Distribution Theory : Test functions and Distributions, Convolutions of Distributions, Fundamental

Sobolev Spaces : Definition and elementary properties of Sobolev spaces, Density results, Extensions,
Traces, Sobolev inequalities, Compactness Theorems.

Weak solutions of second order elliptic boundary value problems: Variational formulation of some
boundary value problems, Lax-Milgram Theorem, Dirichlet and Neumann problems for second order
elliptic operators, Regularity of weak solutions, Maximum principles, Eigenvalue problems.