The aim of this module is to introduce advanced analysis techniques which allow to study partial differential equations and many related problems in mathematics and applications, without the use of numerical and computational tools. The module can loosely be divided in three interconnected parts.

In the first part we study specific functional spaces called Sobolev spaces. Those are a subclass of Lp spaces where it is possible to introduce a notion of weak derivative compatible with integration by parts. The main application of Sobolev spaces is that they provide a suitable class of functions for which divergence form equations are well-posed (i.e. admit a unique solution under appropriate conditions).

The second part of the module is devoted to the general theory of elliptic PDEs in divergence form. We will introduce the notion of weak (non-classical) solutions and learn a basic theory establishing their existence and uniqueness.

The third part is devoted to the basic spectral theory for elliptic PDEs. We will look into such notions as the resolvent operator, eigenvalues and eigenfunctions, and completeness of the latter.

Prerequisite modules: MA3005 Functional and Fourier Analysis

Recommended modules: MA3018 Measure Theory

• ​Demonstrate a thorough understanding of fundamental concepts and techniques of each from the following topics: Lp and Sobolev spaces, Weak derivatives (in the distributional sense), Weak solutions for divergence form PDEs, Existence and uniqueness for divergence form PDEs, Basic spectral theory for self-adjoint elliptic operators.
• Use the above-mentioned concepts and techniques in order to solve various model problems arising in PDEs, including applied one, and establish properties of solutions and relevant differential operators.
• Use the above-mentioned concepts and techniques in order to rigorously prove statements and theorems both of general theoretic and problem specific type in the context of the module material

The module will be delivered predominantly through face-to-face lectures and exercise classes.

The students will also have access to electronic resources such as lecture notes, exercise sheets, e-books, etc. Students are also expected to undertake self-guided study throughout the duration of the module.

• Use of Sobolev spaces to study divergence form PDEs.
• Use of tools of abstract functional analysis in application to elliptic PDEs.
• Use of tools of abstract functional analysis in application to elliptic PDEs.

Transferable Skills:

• Critical thinking.
• Manipulation with abstract notions and rigorous mathematical logic.
• Application of abstract notions and techniques to applied problems.

• The Riesz representation theorem
• The Lax–Milgram theorem
• Sobolev spaces and weak derivatives.
• Weak solutions for elliptic equations in divergence form and example of non-
• existence of classical solutions.
• Self-adjoint elliptic operators.
• Notion of the resolvent.
• Basic spectral theory for self-adjoint elliptic operators.
• The Fredholm alternative.