This course is a continuation of Math 222A from the last semester. Whereas Math 222A concentrated on understanding the simplest examples of PDEs (model linear secondorder PDE's, nonlinear firstorder scalar PDE's) and fundamental tools for studying PDE's (distribution theory, Fourier transform, etc.), Math 222B is where we will begin to learn techniques to deal with linear elliptic and hyperbolic PDE's of greater generality than before and apply them to nonlinear PDEs, which are abundant in mathematics(geometry), science(physics, chemistry, biology etc.), and engineering. We shall focus on those arising from the action principle, which is often how important nonlinear PDEs arise in physics and geometry.
The subjects to be covered are:
 Sobolev spaces: definition, basic properties, density theorems, extension theorems, trace theorems, Sobolev inequalities, RellichKondrachov theorem, PoincarĂ© and Hardy inequalities, duality
 Linear elliptic PDEs: functional analytic tools, basic existence and uniqueness results, elliptic regularity (L2 and Schauder theories), unique continuation, maximum principles, Perron's process
 Linear hyperbolic PDEs: functional analytic tools, basic existence and uniqueness results, oscillatory integrals and dispersive inequalities, vector field method
 Calculus of variations and nonlinear elliptic/hyperbolic PDEs: the action principle, NĂ¶ther's principle, nonlinear hyperbolic and elliptic equations arising from the action principle (time permitting)
