Course tutor: Professor Clement Mouhot

The purpose of this course is to introduce some techniques and methodologies in the mathematical treatment of Partial Dierential Equations (PDE). The theory of PDE is nowadays a huge area of active research, and it goes back to the very birth of mathematical analysis in the 18th and 19th century. It is at a crossroad with physics and many areas of pure and applied mathematics.

The course begins with an introduction to four prototype linear equations:

- Laplace's equation
- the heat equation
- the wave equation
- Schrödinger's equation.

Emphasis will be given to the modern functional analytic techniques relying on the notion of Cauchy problem and estimates rather than explicit solutions, although the interaction with classical methods (e.g. the fundamental solution, Fourier representations) will be discussed. The following basic unifying concepts will be studied: well-posedness, energy estimates, elliptic regularity, characteristics, propagation of singularities, maximum principle. The course will end with a discussion of some of the open problems in PDE.

## Literature

Some lecture notes are available online.

The following textbooks are excellent references:

- Evans, L. C., Partial Dierential Equations, Springer, 2010.
- John, F., Partial Dierential Equations, Springer, 1991.

The following review gives an overview of the eld of PDE: Klainerman, S., Partial Differential Equations, Princeton Companion to Mathematics (editor T. Gowers), Princeton University Press, 2008.

## Additional support

Four examples sheets will be provided and four associated examples classes will be given. There will be a one-hour revision class in the Easter Term.