Nonlinear partial differential equations of mixed elliptic-hyperbolic type in geometry and related areas

Whilst the theory for purely elliptic or purely hyperbolic partial differential equations is relatively well understood, the theory for nonlinear equations of mixed elliptic-hyperbolic type is much less developed. However, equations of mixed elliptic-hyperbolic type have important applications in geometry, as well as a wide range of related areas such as the mathematical study of fluid mechanics, solid mechanics, and elasticity. Developing a greater understanding of equations of mixed type is therefore critical to progress in a number of fields. We illustrate the role of these equations by now giving two examples of problems which can be reduced to the study of equations of mixed elliptic-hyperbolic type, each of which contains a number of open problems and is the subject of current research.
Given a two-dimensional Riemannian manifold, an important problem in differential geometry concerns whether the metric can be realised as an isometric immersion into R-3. This requires us to solve the Gauss-Codazzi equations, which relate the coefficients of the second fundamental form to those of the metric. For surfaces of positive Gauss curvature these can be formulated as an elliptic boundary value problem, whilst for surfaces of negative Gauss curvature we instead have a hyperbolic initial or initial-boundary value problem. In a surface for which the Gauss curvature changes signs, we therefore have to solve an initial-boundary value problem of mixed elliptic-hyperbolic type, for which the available results are much weaker than in the preceding cases. The equivalent problems for higher dimensional manifolds or pseudo-Riemannian metrics can also be studied similarly, and have applications to areas of physics such as relativity.
We also give an example outside geometry, arising from the study of transonic potential flows. We consider a plane shock wave hitting an angled wedge head-on and wish to study the resulting reflection-diffraction pattern, modelled mathematically as a global entropy solution of the two-dimensional Riemann problem for hyperbolic conservation laws. The equations are hyperbolic in the far field and elliptic near the wedge vertex, so again we have a nonlinear PDE of mixed type. We are then interested in the potentially quite complicated structure of the reflection-diffraction patterns, and in how these patterns depend on the wedge angle as well as various physical parameters.
The aim of the research will therefore be to study mixed elliptic-hyperbolic nonlinear PDEs, either in the context of the immersion problem from differential geometry or in a related area of geometry, mechanics or other areas of mathematics in which they arise.
This project falls within the EPSRC Mathematical Analysis research area.