The aim of this module is to introduce advanced analysis techniques which allow to study partial differential equations and many related problems from mathematics and from applications, without the use of numerical and computational tools.  The module consists of 3 different parts.

In the first part we study some specific functional spaces called Sobolev spaces. Those are a subclass of L^P spaces where it is possible to introduce a notion of weak derivative compatible with integration by parts. The main application will be the well-posedness (i.e. existence and uniqueness) for divergence form equations.

The second part of the module is entirely devoted to study the theory of viscosity techniques. These techniques are very different from Sobolev spaces tools, and they are crucial to study nonlinear equations, but also problems from control theory, differential games etc. They do not allow to define derivatives in some weaker sense but they interpret solutions of a wide class of PDEs by testing against smooth functions in a suitable sense (touching from above and from below). The viscosity theory turns out to be crucial in various mathematical problems but also for applications (models from Science and from Finance).

The third part of the module varies each Term and is chosen by the students attending the module from: further topics in Sobolev spaces, application of viscosity theory to degenerate equations, in particular to the evolution by mean curvature flow, control theory and differential games, theory of convex functions.

Even if the module is thought in particular for students with interests in pure mathematics (analysis), the tools presented can be extremely useful for students interested in applications (in both Science and Finance).

Recommended Modules: MA4007 Measure Theory (in particular the Lebesgue integral and the concept of functions coinciding almost everywhere).

The students should gain some knowledge of each of the following topics:

• L^P and Sobolev spaces

  Weak derivatives (in the distributional sense)

• Weak solutions for divergence form PDEs

• Theory of viscosity solutions

• Nonlinear elliptic parabolic PDEs

• Control problems and differential games or mean curvature flows or theory of convex functions (depending on the topic chosen for the third part of the module).

30 fifty-minute lectures

Some handouts will be provided in hard copy or via Learning Central, but students will be expected to take notes of lectures.

Students are also expected to undertake at least 120 hours of private study, including preparation of worked solutions for problem classes.

• Use of Sobolev spaces to study divergence form PDEs.

• Use of viscosity theory.

• Understanding of the rigorous mathematical formulation of control problem/differential games and their connection with the theory of nonlinear PDEs.

  Transferable Skills:

• Knowledge of several advanced analysis techniques and ideas.

• Ability to work with PDEs which are used to describe many different problems from many scientific areas and from finance.

• Ability to use control theory and differential games, which have important applications in engineering and finance.

Formative assessment is carried out in the problem classes.  Feedback to students on their solutions and their progress towards learning outcomes is provided during these classes.  

The summative assessment is the written examination at the end of the module.  This gives students the opportunity to demonstrate their overall achievement of learning outcomes.  It also allows them to give evidence of the higher levels of knowledge and understanding required for above average marks.

The examination paper has a choice of four from five equally weighted questions.

• Overview on L^p spaces.

• Sobolev spaces and weak derivatives.

• Weak solutions for elliptic equations in divergence form and example of non-existence for classical solutions.

• Non- existence of classical solutions for non-divergence form equations: the eikonal equation.

• Viscosity solutions: definitions, properties, Perron’s method and comparison principles.

• Hamilton-Jacobi equations and Hopf-Lax formulas.

• Evolution by mean curvature flow (optional).

• Control system and differential games (optional).

• Theory of convex/semiconvex functions with applications to inf-sup convolutions(optional).

Evans L.C., Partial Differential Equations, American Mathematical Society

Bardi M. and Capuzzo-Dolcetta I., Optimal Control Theory and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Birkhauser, 2008

Brezis, H. Functional analysis, Sobolev spaces and partial differential equations. Springer, 2010.