Math612 - 600: Partial Differential Equations, Spring 2021

Semester log

G.Folland: Introduction to Partial Differential Equations, Mathematical Notes 17, Princeton Univ. Press (see here).
J.Jost: Partial Differential Equations (3rd edition), Springer Graduate Texts in Math 214 (see here).
M. Shubin: Invitation to Partial Differential Equations, AMS Graduate Studies in Mathematics 205 (see here).

• Wednesday, January 20: Begin section 4.3.1 in Evans, Fourier transform, definition, Plancherel's theorem. HW is exercise on page 8 of the notes plus the sheet Homework 1 following page 8.

• Friday, January 22: more on convolution, generalized Young's inequality, properties of the Fourier transform; HW: read the proof of Lemma 2.

• Monday, January 25: application of the Fourier transfrom to compute the fundamental solution of the heat equation; definition of weak derivatives, Sobolev spaces W^k,p(U); HW is exercise on page 2 of the notes, and HW 2 sheet on Schwartz space.

• Wednesday, January 27: examples, elementary properties of W^k,p; HW is page 306, problems 5,6 plus problem on page 8 of the notes.

• Friday, January 29: W^k,p is a Banach space; begin approximation; HW is page 307, problem 9 (first part), read Appendix § C.5.

• Monday, February 1: continue approximation; HW is page 307, problems 9 (second part),11,13,14.

• Wednesday, February 3: go over HW problems, no new material.

• Friday, February 5: Extensions of W^1,p functions; HW is page 308, problems 17,18.
  A recent book on Measure and Integration is by Bercovici, Brown, and Pearcy: click here.

• Monday, February 8: Traces, functions in W^k,p₀; HW is pages 306-307, problems 7,8.

• Wednesday, February 10: finish traces, discuss HW.

• Friday, February 12: Fourier transform method for H^k(Rⁿ), fractional indices; the space H^-1(Rⁿ); HW is exercise on page 4 of the Fourier transfrom methods notes, and problem 20, page 309.

• Monday, February 15 through Friday, February 19: classes cancelled due to inclement weather; make up times to be announced.

• 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 here.

• 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.