Synopsis for C5.1a: Methods of Functional Analysis for Partial Differential Equations

Only available to students who have not offered C5.1a Methods of Functional Analysis for PDEs at Part B.

Level: M-level Method of assessment: Written examination.
Weight: Half-unit (OSS paper code 2A65)

Recommended Prerequisites

Part A Integration. There will be a `Users' Guide to Integration' on the subject website and anyone who has not done Part A Integration can read it up over the summer vacation. In addition some knowledge of functional analysis, in particular Banach spaces (as in B4) and compactness (as in Part A Topology), is useful. We will however recall the relevant definitions as we go along so these prerequisites are not strictly needed.

Overview

The course will introduce some of the modern techniques in partial differential equations that are central to the theoretical and numerical treatments of linear and nonlinear partial differential equations arising in science, geometry and other fields.

It provides valuable background for the Part C courses on Calculus of Variations, Fixed Point Methods for Nonlinear PDEs, and Finite Element Methods.

Learning Outcomes

Students will learn techniques and results about Lebesgue and Sobolev Spaces, distributions and weak derivatives, embedding theorems, traces, weak solution to elliptic PDE's, existence, uniqueness, and smoothness of weak solutions.

Synopsis

Why functional analysis methods are important for PDE's?

Revision of relevant definitions and statements from functional analysis: completeness, seperability, compactness, and duality.

Revision of relevant definitions and statements from Lebesgue integration theory: sequences of measurable functions, Lebesgue and Riesz theorems.

Lebesgue spaces:completeness, dense sets, linear functionals and weak convergence.

Distributions and distributional derivatives.

Sobolev spaces: mollifications and weak derivatives, completeness, Friedrichs inequality, star-shaped domains and dense sets, extension of functions with weak derivatives (with no proof).

Embedding of Sobolev spaces into Lebesgue spaces: Poincare inequality, Reillich-Kondrachov-Sobolev theorems on compactness.

Traces of functions with weak derivatives.

Dirichlet boundary value problems for elliptic PDE's, Fredholm Alternative (uniqueness implies existence).

Smoothness of weak solutions: embedding from Sobolev spaces into space of continuous functions, interior regularity of distributional solutions to elliptic equations with constant coefficients.

1. Lawrence C. Evans, Partial differential equations, (Graduate Studies in Mathematics 2004), American Mathematical Society
2. Elliott H. Lieb and Michael Loss, Analysis, 2nd Edition, (Graduate Studies in Mathematics 2001), American Mathematical Society

Additional Reading

1. E. Kreyszig, Introductory Functional Analysis with Applications, Wiley (revised edition, 1989)
2. P.D. Lax Functional analysis (Wiley-Interscience, New York, 2002).
3. J. Rauch, Partial differential equations, (Springer–Verlag, New York, 1992).