I want to describe a bit more on the history, and a bit of a personal viewpoint on these topics.
Hilbert 19th problem: He asked for analyticity of solutions to elliptic analytic PDEs.
I don't want to present the details of this here. Rather, I want to look at the linearized problem. $$v_t - \Delta v =0, ~y_d = 0.$$ $$ v_t = [\partial_d v], ~ y_d =0.$$
(not Calderon-Zygmund)
DeGiorgi's Conjectore: If $g$ is analytic and $(u,k)$ is a minimizer the $K$ is of class $C^1 \implies K$ is analytic.
Euler-Lagrange Equations: $$ \Delta u^{\pm} = g^{\pm} (x, u^{\pm}, Du^{\pm}), ~ \Omega \backslash K.$$ $$ \frac{\partial u^{\pm} }{\partial \nu} = b^{\pm} (x, u^+, u^-)$$
$k$ is the mean curvature.
Theorem (Leoni, Marini 2005): The conjecture is true.
proof: We want to straighten out $K$ but there's noting obvious in the problem proving a method to do this. Draws some pictures and generates an elliptic system to which work by Morrey applies.
You strive to minimize the Dirichlet energy subject to given boundary conditions above an obstacle. Signorini problem. This is mimicked by the thin obstacle problem.
$u: B_1^+ \rightarrow \mathbb{R}$
$u|_{x_d = 0} \geq 0.$
$ \int |\nabla u |^2 dx \leq \int |\nabla v|^2 dx, ~ \forall ~ v: v =u ~{\mbox{on}}~ |x| =1, ~ x_d >0$
$ v \geq 0$ at $x_d = 0$.
At the level of the differential equation
$\Delta u - 0, ~ B_1^+$
$ u \cdot \partial_{x_d} u = 0$ on $x_d = 0$.
$ u\geq 0, \partial_{x_d} u \leq 0.$
Fact: $u \in C^{1,1/2}.$
Contact set: $u=0$ on $x_d = 0$.
Free boundary: $\partial \{ u > 0 \}.$
We can blow-up at the free boundary to explore regularity.
$$ u^\lambda (x) = \frac{u (\lambda x)}{{\| u (\lambda x) \|}_{L^2 (B_1^+ )}}$$$u^\lambda (x)$ has a limit $u^\infty$ as $\lambda \rightarrow \infty.$
$u^\infty$ is a homogeneous solution.
Theorem (Koch-Petrosyan-Shi): Regular part of the free boundary is analytic.
This problem is different than what we've discussed previously. This is codimension 2.
Choice of coordinates, Hodograph transform,