General Prerequisites: A4 Integration. There will be a `Users' Guide to Integration' on the subject website and anyone who has not learned Lebesgue Integration can read it up over the summer vacation. In addition some knowledge of functional analysis, in particular Banach spaces and compactness, is useful.
Course 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 \(L^p\) and Sobolev \(W^{1,p}\) Spaces, distributions and weak derivatives, embedding theorems, traces, weak solution to elliptic PDE's, existence, uniqueness, and smoothness of weak solutions.
Course Synopsis: Why functional analysis methods are important for PDE's?
Revision of relevant definitions and statements from functional analysis: completeness, separability, compactness, and duality.
Revision of relevant definitions and statements from Lebesgue integration theory: sequences of measurable functions, Lebesgue and Riesz theorems.
Lebesgue spaces $L^p$: completeness, dense sets, linear functionals and weak convergence.
Distributions and distributional derivatives.
Sobolev spaces $W^{1,p}$: mollifications and weak derivatives, completeness, Friedrichs inequality, star-shaped domains and dense sets, extension of functions with weak derivatives.
Embedding of Sobolev spaces into Lebesgue spaces: Poincaré 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), variational method, spectrum of elliptic differential operators under Dirichlet boundary conditions.
Smoothness of weak solutions: embedding from Sobolev spaces into spaces of Hölder continuous functions, regularity of distributional solutions to elliptic equations with continuous coefficients. Strong solutions to the Dirichlet problem for elliptic differential operators.
