** Lecturer:**. Karsten Matthies. Email: k.matthies --at -- bath.ac.uk Phone: (01225) 383858

This course is run as part of the Mathematics Taught Course Centre organised in collaboration with Bristol, Imperial, Oxford and Warwick. This is an introductory course for research students in pure or applied mathematics who have some basic familiarity with Lebesgue integration and Functional Analysis. (Follow the links below for a summary of background results that will be used in this course.)

**Topics:**.

Distributions and their partial derivatives, Distributional derivatives of functions, Weak Solutions of partial differential equations, Sobolev spaces, Convolution and mollification, Density of smooth functions in Sobolev spaces, Sobolev Inequality, The Poincare Inequality, Morrey's Inequality, Rellich-Kondrachov compact embedding theorem, Simple examples on existence of weak solutions, Boundary values of Sobolev functions, Use of weak compactness methods.

**Bibliography:**

1) L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics 19, American Mathematical Society, 2nd Edn. 2010.

2) R. A. Adams & J.J.F. Fournier, Sobolev Spaces, Elsevier, 2nd Edn. 2003.

3) V. G. Maz'ya, Sobolev spaces, Springer, 2nd Edn. 2011. (For reference.)

4) H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, 2011.

5) G. Leoni, A First Course in Sobolev Spaces, Graduate Studies in Mathematics 181, American Mathematical Society, 2nd Edn., 2017.

**Lecture notes:**

Background notes: Lebesgue integration and Functional Analysis

Chapter 1: Preliminaries, Chapter 2: Sobolev Spaces , Chapter 3: Regularisation and approximation, Chapter 4: Embeddings on Smooth Bounded Domains

**Screen notes:**

Lecture 1, Lecture 2, Lecture 3, Lecture 4, Lecture 5 (this year's found again), Lecture 6, Lecture 7

**Example sheets:**

Sheet 1 , Sheet 2 , Sheet 3, Sheet 4, Sheet 5, Sheet 6

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