Elliptic partial differential equations: theory and applications

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Partial differential equations (pde's) lie at the heart of many mathematical models which attempt to describe various physical phenomena in the real world.    Because of this understanding pde's is of the utmost importance.   Mathematically pde's is a very broad area and generally one concentrates on certain classes of pde's.   My area of expertise is in the area of elliptic partial differential equations and in particular in the area of semilinear and quasilinear elliptic pde's. I propose to examine various elliptic  equations which are not variational in structure.  In the case of variational pde's one can attach an energy functional whose critical points correspond to solutions of the pde.  Often this allows one to use the very well developed machinery of variational calculus, for instance min-max techniques, to obtain existence and multiplicity results.  When the equation is no longer variational one does not have this variational machinery available and one must utilize other methods, for instance one can try topological methods.   In addition I propose to study some elliptic problems which are variational in nature but are supercritical, which in general, does not allow for a variational approach.    These problems range in difficulty from fairly straightforward to quite challenging.    Some of these problems are very  suitable for a masters student and do not require a lot of background.