# Boundary Harnack inequality

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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

- 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

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