Variational Problems on Arbitrary Sets

Let $E$ be an {\em{arbitrary}} subset of $\mathbb{R}^n$. Given functions $f: E \rightarrow \mathbb{R}$ and $g: \mathbb{R}^n \rightarrow \mathbb{R}$, the classical obstacle problem asks for a minimizer of the energy functional $\mathcal{E}(F) = \int_{\mathbb{R}^n} |\nabla F|^2$ subject to the following two constraints: (1) $F = f$ on $E$ and (2) $F \geq g $ on $\mathbb{R}^n$. In this talk, we will discuss how to use extension theory to construct the solutions directly. We will also explain several recent results that will help lay the foundation for building a complete theory revolving around the belief that any variational problems that can be solved using PDE theory can also be dealt with using extension theory. This talk draws on joint work with C. Fefferman and A. Israel.