Lecture 11, Calculus of Variations and the Mountain Pass Theorem

On December 19, we started a brief excursion into nonlinear problems.

1. We had a crash course on variational methods, covering portions of Chapter 8 in Evans, including the essence of the so-called direct method of the calculus of variations:
   • First variation of a functional, Euler–Lagrange equations;
   • Simple examples;
   • Second variation of a functional;
   • Weak convergence in Sobolev spaces, the notion of sequentially weakly lower semicontinuous (for short: swlsc) functionals;
   • Coercivity; all functionals with lagrangians that are convex in the gradient variable are swlsc (Theorem 1 in Evans, Section 8.2.2);
   • Existence of minimizers (Theorem 2 in Evans, Section 8.2.2) and, as a corollary, existence of weak solutions of Euler–Lagrange equations.

2. We discussed the Mountain Pass Theorem, following Evans, Section 8.5.1 (with emphasis on the geometric interpretation of the proof of the Deformation Theorem - if you are ever going to read this part of Evans, please try and make a good picture of the graph of a functional on a Hilbert space to explain the whole notation of Evans, Sec. 8.5.1(a) and the statement of Theorem 1 there).

3. Finally, an application to semilinear elliptic equations was announced. To understand it with all the details, you should either read Evans, Sec. 8.5.2, or solve the appropriate problems that are posted on my page.