Boundary Harnack inequality and Category:Quasilinear equations: Difference between pages

From nonlocal pde
(Difference between pages)
Jump to navigation Jump to search
imported>RayAYang
(add a bit)
 
imported>Nestor
No edit summary
 
Line 1: Line 1:
The '''Boundary Harnack Inequality''' is a name given to two related statements for nonnegative functions $u$ which are solutions of elliptic equations.
A quasilinear equation is one that is linear in all but the terms involving the highest order derivatives (whether they are of fractional order or not). For instance, the following equations are quasilinear


The first result, also known as '''Carleson's estimate''', says that for non-negative solutions, their values in a neighborhood of the (suitably smooth) boundary are bounded in terms of the value at some interior point. Let $u$ be a non-negative solution of an elliptic equation $Lu = 0$ on some domain $\Omega \subset \mathbb{R}^n$, such that $u = 0$ on $B_r(x_0) \cap \partial \Omega$, where $x_0$ lies on the boundary $\partial \Omega$, and $x'$ is some other point lying within $B_\frac{r}{2}(x_0) \cap \Omega$. Then, inside $B_\frac{r}{2}(x_0) \cap \Omega$, there exists a constant $M > 0$ such that $u(x) \leq M u(x')$.
\[ \mbox{div} \left ( \frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right ) = 0 \]
 
\[ u_t = \mbox{div} \left ( u^p \nabla u\right ) \]
The second result, also known as the '''boundary comparison estimate''', says that two non-negative solutions which are zero on some portion of the boundary, have a Holder continuous ratio with respect to each other in some neighborhood of the boundary. That is, let $ Lu = Lv = 0$ inside some domain $\Omega$ with smooth boundary, with $u,v \geq 0$, and $u = v = 0$ along $B_r(x_0) \cap \partial \Omega$ for some $x_0 \in \partial \Omega$. Then the ratio $\frac{u}{v}$ lies in the Holder class $C^\alpha (B_\frac{r}{2}(x_0))$.
\[ (-\Delta)^{s} u +H(x,u,\nabla u)=0 (2s>1) \]

Revision as of 17:09, 3 June 2011

A quasilinear equation is one that is linear in all but the terms involving the highest order derivatives (whether they are of fractional order or not). For instance, the following equations are quasilinear

\[ \mbox{div} \left ( \frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right ) = 0 \] \[ u_t = \mbox{div} \left ( u^p \nabla u\right ) \] \[ (-\Delta)^{s} u +H(x,u,\nabla u)=0 (2s>1) \]