Iterative improvement of oscillation

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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 $\alpha = \log(1-\delta)/\log(\rho)$.