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
Brief Biography
Melih Ucer is a postdoctoral researcher in the Mean-Field Games (MFG) group of Prof. Diogo Gomes at KAUST. He obtained his bachelor's degree from MIT in physics and PhD degree from Bilkent University in mathematics, where he did research on topology of algebraic varieties. Since he joined KAUST, he has been primarily working on the weak solution concepts to MFG equations. In addition, being a former IMO medalist, he is still active in training olympiad students.