Monday, February 22: finish H-1(Rn); extension for higher order Sobolev spaces; HW is the exercise on Vandermonde determinants on page 3.
Wednesday, February 24: the Sobolev lemma, the Rellich lemma; HW as assigned in notes. Exam 1 will be distributed on Thursday morning, due Monday, March 1 before class.
Friday, February26: finish Rellich lemma; remaining class time to work on exam.
Monday, March1: Exam 1 due; Poincaré inequality; begin Chapter 6, second order elliptic equations, ellipticity, weak solutions.
Wednesday, March 3: more on weak solutions, Lax-Milgram theorem; HW as assigned in class plus problem 2, page 365.
Friday, March 5: Go over Exam 1 problems.
Monday, March 8: Energy estimates, alternative version of Poincaré inequality, L+μ as an isomorphism from H10(U) onto H-1(U); HW is problems 2, 3 on pages 365-366.
Wednesday, March 10: (more) examples of weak derivative in L2 vs derivative in H-1; spectrum of L.
Friday, March 12: Fredholm alternative; HW is problems 4,5 on page 366.
Monday, March 15: local regularity for strongly elliptic second order operators, Friedrichs' lemma; HW is to study proof of Friedrichs' lemma plus problem 11, page 367.
Wednesday, March 17: higher order interior regularity; HW is problem 7, page 366.
Thursday, March 18: redefined day, students attend Friday classes; no new material, discuss HW problems.
Monday, March 22: begin boundary regularity, H2(U) regularity; HW is exercise on page 5b of the notes (use Rellich lemma to prove interpolation inequality).
Wednesday, March 24: higher boundary regularity.
Friday, March 26: problems with inhomogeneous boundary condition, existence and regularity.
Monday, March 29: class cancelled due to technical difficulties; Exam 2 due today.
Wednesday, March 31: weak maximum principles for (uniformly) elliptic second order operators; begin Hopf lemma.
Friday, April 2: reading day, no classes.
Monday, April 5: Discuss Exam 2 problems.
Wednesday, April 7: Hopf lemma, strong maximum principle; HW is problem 9 on page 367.
Here is an extensive resource for elliptic second order PDEs: Gilbarg/Trudinger, Elliptic Partial Differential Equations of Second Order. Free download is
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Friday, April 9: eigenvalues and eigenfunctions, eigenvalues of symmetric elliptic operators.
Monday, April 12: finish eigenvalues and eigenfunctions; HW is pages 368-369, problems 13, 15.
Wednesday, April 14: function spaces involving time; begin second order parabolic equations.
Friday, April 16: weak solutions, Galerkin approximation.
Monday, April 19: energy estimates for Galerkin solutions, discuss HW.
Wednesday, April 21: existence of weak solutions; HW is problem 5 on page 447, exercise given in notes.
Friday, April 23: discuss integration by parts in function spaces involving time.
Monday, April 26: begin regularity for second order parabolic equations, formal discussion; HW is pages 446-447, problems 4, 6.
Wednesday, April 28: Discuss HW problems; end of classes for this semester; Exam 3 given out after class, due Monday, May 3, 11:00am.