Template:Citation/patent and Boundary Harnack inequality: Difference between pages
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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== | |||
*{{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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Revision as of 09:44, 9 July 2011
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
- Caffarelli, Luis; Salsa, Sandro (2005), A geometric approach to free boundary problems, Graduate Studies in Mathematics, 68, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-3784-9, Chapter 11