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.