Syllabus for
Math 554. Linear Anal & Part Diff Eq
Recommended texts
1. G. B. Folland, Real Analysis
2. L. C. Evans, Partial Differential Equations
3. P. D. Lax, Functional Analysis
4. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations
5. J. Hunter and B. Nachtergaele, Applied Analysis
During the run of the course we will review the basic theorems of Functional Analysis. Examples include Hahn-Banach, Banach-Steinhaus, the open mapping theorem, the closed graph theorem, Riesz representation.
Chapter 1 (3 hours). Banach Fixed Point Theorem and applications to differential and integral equations.
Chapter 2 (8 hours). The theory of L^p spaces. Completeness, Duality, Reflexivity, Convolutions and Molification along with the basic inequalities that we use in PDEs. Riesz-Thorin interpolation theorem.
Chapter 3 (7 hours). Introduction to the spectral theory of bounded linear operators. The spectrum of compact operators (Hilbert-Schmidt), Fredholm Alternative and applications to Layer Potentials.
Chapter 4 (3 hours). Introduction to the class of unbounded operators. Criteria for self-adjointness.
Chapter 5 (7 hours). Distributions and Fourier Transform on R^n. L^2-based Sobolev spaces, Fundamental Solutions, Green’s Functions. Heat, Schrödinger or Linear Wave equation on R^n by inverting the Fourier transform.
Chapter 6 (9 hours). Weak derivatives and Sobolev spaces on bounded domains (W^kp). Embedding Theorems, Rellich-Kondrachov Theorem. Elliptic regularity. Lax-Milgram and applications to linear evolution equations (non constant coefficients). Galerkin approximation. Examples on Elliptic, Parabolic and Hyperbolic equations.
Chapter 7 (5 hours). Introduction to Semigroup Theory. Hille-Yosida Theorem and its applications to linear equations (Heat, Wave, Schrödinger). Examples of Nonlinear Evolution equations through the applications of the abstract semigroup theory on Sobolev spaces.
Prerequisites
Math 447, Math 489, or consent of instructor. Math 540 would be useful, but is not required.
Grading Policy
We suggest regular homeworks (which includes some of the skipped proofs) and a takehome final exam.