Title of Project:
Viscosity Solutions and Nonlinear Partial Differential Equations

Code: FD2122B

Supervisor: Dr. F. Dragoni

Project description:
Nonlinear partial differential equations (PDEs) describe many phenomena
in sciences and economics, e.g. the value of an optimal control problem,
minimal surfaces, porous media, conservation laws, etc. Solutions to these
equations are often not sufficiently regular in order to write all the derivatives
appearing in the equation (e.g. they can be just continuous and not
differentiable). So the solutions have to be understood in a suitable generalized
sense. We consider the so-called viscosity solutions which consist
in using sufficiently smooth test functions to approximate from above and
below a unique continuous solution for a very large class of first-order and
second-order PDEs.
A project would consist of an introduction to the general theory of viscosity
solutions and applications to a specific nonlinear PDE (e.g. Hamilton-Jacobi
eq.s, infinite-Laplace eq., p-Laplace eq., etc.). One possibility is to investigate
the relation between viscosity solutions and control theory or games
theory. Another possibility is to focus on degenerate elliptic/parabolic equations
(e.g. the evolution by mean curvature flow) or the relations with convexity.
If required, it will be possible to deal with more geometric equations
(space-dependent) and metric formulas. One possibility is to investigate the
relation between viscosity solutions and control theory or games theory.

Project offered a double module, single module, or both:

Prerequisite 3rdd year modules:
The following classes may be helpful:
MA3303 Theoretical and Computational PDEs

Number of students who can be supervised on this project:
1