Instructor: Malgorzata Peszynska.
http://math.oregonstate.edu/~mpesz/teaching/627_F22/
This class will develop the theory of variational methods in Hilbert spaces for elliptic Partial Differential Equations. This theory forms the background for the qualitative study of many types of PDEs as well as for their approximation with finite element method. For evolution problems, we will proceed via semigroup theory. These techniques will allow a deeper study of the PDE models discussed in MTH 621-2-3 including the flow and mechanics models. We will also cover some nonlinear problems including variational inequalities for problems under constraints (based on some basics of convex analysis to be developed), and additional topics based on the variational techniques to be developed.
Prerequisites: Material covered in MTH 621-2 or similar, and strong interest in the subject. [I will develop background and provide additional resources as needed].
Text: we will likely use portions of the text by R.E. Showalter, "Hilbert Space Methods for Partial Differential Equations", available online, as well as other materials to be circulated; Zeidler, "Nonlinear Functional Analysis and Applications, I-IV"; available through Springer link; Oden, "Applied Functional Analysis", handouts available as needed; my notes on various problems, to be made available as needed.
Thanks for your interest. Go PDEs!

Schedule: [resources] {assignments}

    == Week 0 [Showalter.I]
  1. 9/21 Class overview: review basic notions of vector spaces and inner product spaces.
  2. 9/23 Convex functions and sets. L^p spaces. Linear operators in L(V,W). Continuous (bounded) linear operators in {\em L}(V,W). Functionals L(V,R). Algebraic dual V^*. Dual space V'.

    3. == Week 1
  3. 9/26 Hilbert spaces; subspaces and ortogonal decomposition. [Showalter.I]
  4. 9/28 Riesz representation theorem (two proofs including "standard" and one on minimization principle in a reflexive Banach space in a convex and closed subset of a convex l.s.c function)
  5. 9/30 {Class activity on [Showalter.I.1-18 minus completion]}

    == Week 2
  6. 10/3 [Showalter, II]. Distributional derivatives.Poincare-Friedrichs inequality (different proofs).
  7. 10/5 Weak derivatives vs derivatives in L^2. Sobolev spaces H^m and {\mathcal H^m}. Meyers-Serrin result.
  8. 10/7 Equivalent norm/inner product on H_0^1 via PF inequality. Weak formulation of a Dirichlet BVP, and E/U via Riesz for the sel f-adjoint case.

    9. == Week 4
  9. 10/10 Weak formulation of Neumann problem. Partial equivalence of classicial and weak formulations. D as an unbounded operator, and as an operator bounded from below.
  10. 10/12 Bilinear forms which are continuous and coercive.
  11. 10/14 {Class activity on weak and distributional derivatives, and Sobolev spaces}.

    12. == Week 5
  12. 10/17 Lax Milgram Theorem: 2 proofs.
  13. 10/19 Lax-Milgram Theorem: other proofs. Demystifying variational inequalities. Setting up BVPs in weak form so Lax-Milgram can be applied
  14. 10/21 More BVPs in weak form. (Partial) equivalence between classical and weak formulations.

    15. == Week 6
  15. 10/24: Trace theorem (guest lecture)
  16. 10/26: Sobolev embedding theorem (guest lecture)
  17. 10/28: Class activity on coercivity and continuity, as well as Robin b.conditions.

    18. == Week 7
  18. 10/31: Wrap-up different (inhomogeneous) BVPs [III.1-5]. Regularity [III.6]
  19. 11/2: Class CANCELLED due to illness.
  20. 11/4: Elliptic Variational Inequalities: proof of Lax-M via Stampacchcia proof; proof of Stampacchia theorem. Examples. [V.3]

    21. == Week 8
  21. 11/7: Finite Elements: introduction for conforming spaces. [V.5]
  22. 11/9: FE: energy error estimates for linear Galerkin; examples in 1d.
  23. 11/11: NO CLASS (Veteran's day)

    24. == Week 9
  24. 11/14 FE: finite element spaces; interpolation.
  25. 11/16 FE: building conforming vs non-conforming spaces.
  26. 11/18 FE: errors in non-energy norm.

    27. == Week 10
  27. 11/21: Mixed setting for (some) elliptic PDEs. Introduction and linear algebra examples.
  28. 11/23: Theory of mixed approaches ([Boffi, Brezzi, Fortin]).
  29. 11/25: no class Thanksgiving holiday)

    30. == Week 11
  30. 11/28: Connect mixed approaches via BNB1 and BNB2 conditions: Lax milgram as a special case. Semigroup approach for evolution PDEs: introduction.[Chapter IV.]
  31. 11/30: Continue semigroup theory for parabolic and hyperbolic problems. Calculate adjoints.
  32. 12/2: Wrap-up: semigroup.