The textbook for this course is: L.C. Evans, Partial Differential Equations, 2nd Edition .

However, we will often go beyond or skip materials in the textbook. Useful references will be posted below:

• Lectures notes from 222A: pdf. In addition to Evans's book, we will use Section 14 of these lecture notes for Sobolev spaces.

All course announcements will be posted on the class bCourses page: link.

Time: TuTh 12:30pm - 2pm

Location: Evans 31

Online classroom: Until 1/30, we will have online lectures at the following Zoom link:
https://berkeley.zoom.us/j/95642667565?pwd=TUhVYmNVdTE3Z0kvMEpLWlg0ZFNSUT09

Time: TuTh 5pm - 6pm or by appointment

Location: Evans 887

Online meeting room: Until 1/30, we will have online office hours at the same Zoom link as lectures:
https://berkeley.zoom.us/j/95642667565?pwd=TUhVYmNVdTE3Z0kvMEpLWlg0ZFNSUT09

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 second-order PDE's, nonlinear first-order 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, Rellich-Kondrachov theorem, Poincaré 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, Nöther's principle, nonlinear hyperbolic and elliptic equations arising from the action principle (time permitting)