Monotone Operator methods for proving existence of solutions to Dirichlet-type problems
I will discuss several standard techniques for proving existence of solutions to boundary value problems involving second-order elliptic PDE.
Overview
The techniques that I will discuss are bound by the common theme of formulating a boundary value problem on a suitable Sobolev space. I will begin with the Lax-Milgram theorem and Calculus of Variations, and then proceed to the Galerkin method with monotone-like operators. I will emphasize that, in each of these techniques, the key component is a certain one-sided inequality condition: positivity of a bilinear form, convexity of a functional or monotonicity of an operator.
Presenters
Breif Biography
Dr. Melih Üçer received his B.S. in physics from MIT and Ph.D. in mathematics from Bilkent University, where he did research on geometric topology. He has been a postdoc at KAUST in Prof. Diogo Gomes' group since 2023, working primarily on existence and regularity of weak solutions to mean-field game equations. He is also active in training students for mathematical olympiads.