MATH 7530 Partial Differential Equations I

Classical weak and strong maximum principles for 2nd order elliptic and parabolic equations, Hopf boundary point lemma, and their applications. Sobolev spaces, weak derivatives, approximation, density theorem, Sobolev inequalities, Kondrachov compact imbedding. L2 theory for second order elliptic equations, existence via Lax-Milgram Theorem, Fredholm alternative, a brief introduction to L2 estimates, Harnack inequality, eigenexpansion. L2 theory for second order parabolic and hyperbolic equations, existence via Galerkin method, uniqueness and regularity via energy method. Semigroup theory applied to second order parabolic and hyperbolic equations. A brief introduction to elliptic and parabolic regularity theory, the Lp and Schauder estimates. Nonlinear elliptic equations, variational methods, method of upper and lower solutions, fixed point method, bifurcation method. Nonlinear parabolic equations, global existence, stability of steady states, traveling wave solutions. Conservation laws, Rankine-Hugoniot jump condition, uniqueness issue, entropy condition, Riemann problem for Burger's equation, p-systems.