Variational and topological methods for nonlinear partial differential equations

In this course we study nonlinear differential equations, using analytical, geometrical and topological ideas.  Nonlinear partial differential equations and dynamical systems arise in a variety of applications, but
they are also studied for their intrinsic mathematical interest.  While the nonlinearity of the problem prevents direct calculation of solutions, several beautiful techniques have been developed to find solutions
nevertheless.

First, instead of looking for solutions of the differential equations directly, one reformulates the problem as one where we try to find the minimum of a function(al), or more generally, a critical point (e.g. a mountain pass or saddle).  If such a reformulation is possible, it turns
out that it is usually much easier to discover such critical points, than to attack the differential equation(s) head on.  Such a (so-called) variational approach requires both an investigation of the geometry of critical points of the functional, as well as a carefully chosen functional analytic setting in which to carry out the proofs. Examples include energy minimizing configurations of elastic materials or soap bubbles, semilinear elliptic partial differential equations, and geodesics on Riemannian manifolds.

Second, if a variational structure is not available, one can often adopt an alternative approach to the problem by means of a fixed point construction,
where fixed points of a suitably chosen map correspond to solutions of the differential equation(s) under consideration.  Again, a functional analytic
framework is necessary to make this topological idea work in infinite dimensional (function) spaces (in finite dimensions one may think of Brouwer's fixed point theorem, but also Newton's iteration method for finding zeros).

In the course we will develop general theory, but we will also discuss plenty of examples and applications.  Care will be taken throughout the course to to provide the variational and topological ideas in conjunction with the technical machinery required.