PhD seminar

Introduction to Lyapunov-Schmidt reduction methods for PDEs

The seminar will take place at Zoom. Passcode: 478793

Abstract:

In the study of PDEs, and in particular non-linear PDEs, one often relies on the Implicit Function Theorem. For example, in the method of continuity, which involves the embedding of the given problem in a parametrized family of problems, the Implicit Function Theorem is applied to establish solvability near a parameter point where the solution is known to exist. While the method of continuity is very powerful and has been applied to solve many famous PDEs such as geometric PDEs of mean curvature type and Monge-Ampere equations, a serious restriction comes from the fact that the Implicit Function Theorem requires the linearization of the differential operator to be continuously invertible.  In this talk, we will discuss the so-called finite dimensional Lyapunov-Schmidt reduction method, which is a method for solving problems where the linearization is merely Fredholm, i.e. it has finite kernel and its range has finite codimension. As an application of this method, we will solve the non-linear Schrödinger equation.