Iterative improvement of oscillation and Fully nonlinear elliptic equations: Difference between pages

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(Created page with "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...")
 
imported>Luis
(Created page with "A fully nonlinear elliptic equation is an expression of the form \[ F(D^2u, \nabla u, u, x) = 0 \text{ in } \Omega.\] The function $F$ is supposed to satisfy the following two b...")
 
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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.
A fully nonlinear elliptic equation is an expression of the form
\[ F(D^2u, \nabla u, u, x) = 0 \text{ in } \Omega.\]


= Main scaling assumption =
The function $F$ is supposed to satisfy the following two basic assumptions
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
* If $A \geq B$, $F(A,p,u,x) \geq F(B,p,u,x)$ for any values of $p \in \R^n$, $u\in \R$ and $x \in \Omega$.
\[ u_r(x) = \lambda u(rx),\]
* If $u \geq v$, $F(A,p,u,x) \leq F(A,p,v,x)$ for any values of $A \in \R^{n \times n}$, $p \in \R^n$ and $x \in \Omega$.
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 $\alpha = \log(1-\delta)/\log(\rho)$.

Revision as of 16:51, 8 April 2015

A fully nonlinear elliptic equation is an expression of the form \[ F(D^2u, \nabla u, u, x) = 0 \text{ in } \Omega.\]

The function $F$ is supposed to satisfy the following two basic assumptions

  • If $A \geq B$, $F(A,p,u,x) \geq F(B,p,u,x)$ for any values of $p \in \R^n$, $u\in \R$ and $x \in \Omega$.
  • If $u \geq v$, $F(A,p,u,x) \leq F(A,p,v,x)$ for any values of $A \in \R^{n \times n}$, $p \in \R^n$ and $x \in \Omega$.
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