Synopsis

The aim of this course is to provide the prerequisities on L^p and Sobolev spaces needed for the courses Calculus of Variations, and Finite elements for PDEs and possibly other courses (in particular, it will be recommended background for the Solid Mechanics course). The course will cover a lot of material in the style of a methods course, with plenty of examples and the main ideas and results described carefully and precisely, but with only illustrative proofs given.

Brief review of background analysis. Banach spaces and their duals. The Lebesgue integral, Lebesgue measure and L^p spaces. The dual space of L^p for 1 < p < \infty. Hausdorff measure. (3 lectures)

Weak and weak* convergence. Proof that if f : R -> R is such that u -> f(u) is weakly continuous from L^p to L^p then f is affine. Young measures as a tool for characterizing weak limits of f(u) for nonlinear f. (3 lectures).

Sobolev spaces: Definitions and examples. Mollifiers and approximation by smooth functions. The fundamental theorem of calculus using weak derivatives. Extension and embedding theorems. Traces. Poincarés inequality. (7 lectures)

Elementary theory of distributions.(1 lecture)

Weak formulation of second-order linear elliptic boundary value problems. The Lax-Milgram lemma. Existence and uniqueness of weak solutions. (2 lectures)

Reading List

• W P Ziemer, Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation, Springer, 1989

• L C Evans and R F Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, 1992

• R A Adams Sobolev Spaces, Academic Press, 1975

• H Brezis Analyse Fonctionelle, Masson, 1983

• V G Mazya Sobolev Spaces, Springer, 1985

• E H Lieb and M Loss Analysis AMS, Graduate Studies in Mathematics, Vol,14, AMS, Providence, 1997