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