Topics to be covered

Introduction to Sobolev Spaces (Quick review)

Ø  Multivariable Calculus: Derivative as a linear map, examples, computing the derivative in terms of partial derivatives, Taylor’s theorem up to second order and applications to maxima-minima, Inverse function theorem, Implicit function theorem.

Ø  Advanced Functional Analysis: c, Compactness and weak convergence, Riesz Representation Theorem, Hilbert spaces and Banach Spaces.

Ø  Sobolev Spaces: Test functions and distributions, Sobolev spaces; concept of weak derivative, Compactness and Embeddings, Trace theorems;

Classical Partial Differential Equations:

Ø  Laplace equation: Harmonic functions, maximum principle, fundamental solution, Green’s function and Energy methods.

Ø  Heat equation: Fundamental solution, Mean-value formula, Weak and strong maximum principles for the heat equation, Energy methods

Ø  Wave equation: Solution by spherical means, d’Alembert’s solution, Non homogeneous problem, Energy method

Weak Solution of Linear Partial Differential Equations:

Ø  Existence of Weak Solution of Elliptic equation, Lax-Milgram Theorem, Energy estimates, Fredholm alternative, Regularity

Ø  Existence and uniqueness of Weak Solution of parabolic equation, Regularity

Ø  Existence and uniqueness of weak solution of hyperbolic equation, Regularity