View source for Iterative improvement of oscillation

This is a generic technique which is used to show that solutions to some equations are Holder continuous. The technique consist in proving iteratively that the oscillation of the function in a ball of radius $r$ (for elliptic equations) or a parabolic cylinder (for parabolic equations) has a certain decay as $r \to 0$ and then obtain a Holder continuity result from it.

= Main scaling assumption =
In order for the technique to work, we need to start with a solution to a scale invariant equation or class of equations. That is, we have a function $u : B_1 \to \R$ such that, the scaled functions
\[ u_r(x) = \lambda u(rx),\]
satisfy some convenient equation for all $r<1$ and $\lambda>0$. The equation can depend on $r$, as long as the assumptions on it do not deteriorate as $r \to 0$.

= What we need to prove =
== Main lemma ==

The main step is to prove that there exists a radius $\rho>0$ and $\delta>0$ so that for any solution $u$ such that $\textrm{osc}_{B_1} u \leq 1$ then $\textrm{osc}_{B_\rho} u \leq 1-\delta$.

Alternatively, for parabolic equations, we would have to prove that if 
\[ \textrm{osc}_{B_1 \times [-1,0]} u \leq 1 \ \text{ then } \ \textrm{osc}_{B_\rho \times [-\rho^2,0]} u \leq 1-\delta.\]

== How it works ==
Iterating a scaled version of the main lemma mentioned above, we get that for all integers $k>0$,
\[ \textrm{osc}_{B_{\rho^k}} u \leq (1-\delta)^k.\]
This implies that $u$ is $C^\alpha$ at the origin for $\alpha = \log(1-\delta)/\log(\rho)$.