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MTH-207.4 Advanced PDE

Review of harmonic functions, Extension of maximum principles to 2nd order elliptic equations, existence via sub-super solutions.

A priori estimates of Schauder, existence via fixed point method.

Existence via variational methods: direct minimization and constrained minimization.

Evolution equations: existence via semigroup theory,

Well-posedness of Cauchy Problem for strictly hyperbolic systems, Initial and Initial- boundary value problems for symmetric hyperbolic systems, and linear second order systems,

Nonlinear Hyperbolic systems- Asymptotic solutions of oscillatory initial value problems (Lax Theory)

Scalar conservation laws (Hopf-Lax formula). Systems of hyperbolic conservation laws- 2 x 2 Lax theory. General Case - introduction to Glimm's theory.