Partial Differential Equations II

Course content

Sobolev spaces: Weak Derivatives, Approximation by Smooth functions, Extentions, Traces, Sobolev Inequalities, The Space H^-1. Second-order Elliptic Equations: Weak Solutions, Lax-Milgram Teorem, Energy estimates, Fredholm Alternative, Regularity, Maximum Principles, Eigenvalues and Eigenfunctions. Linear Evolution Equations: Second-order Parabolic equations, (Weak solutions, Regularity, Maximum principle), Second-order Hyperbolic Equations (Weak Solutions, Regularity, Propagation of disturbances), Hyperbolic Systems of First-order Equations, Semigroup theory.