Boundary Harnack inequality: Difference between revisions

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==Further reading==
==Further reading==
*{{Citation | last1=Caffarelli | first1=Luis | last2=Salsa | first2=Sandro | title=A geometric approach to free boundary problems | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Graduate Studies in Mathematics | isbn=978-0-8218-3784-9 | year=2005 | volume=68}}, Chapter 11
*{{Citation | last1=Caffarelli | first1=Luis | last2=Salsa | first2=Sandro | title=A geometric approach to free boundary problems | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Graduate Studies in Mathematics | isbn=978-0-8218-3784-9 | year=2005 | volume=68}}, Chapter 11
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Latest revision as of 21:34, 5 February 2012

The Boundary Harnack Inequality is a name given to two related statements for nonnegative functions $u$ which are solutions of elliptic equations.

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 $u(x) \leq M u(x')$.

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 $C^\alpha (B_\frac{r}{2}(x_0))$.

Further reading

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