# Laplacian Boundary Value Problem

Given a domain $M \subset \mathbb{R}^2$ and a function $f : \partial M \rightarrow \mathbb{R}$, let $\omega$ solve the boundary value problem:
$$\Delta \omega = 0 \text{ in } M \\ \omega = f \text{ in } \partial M$$
I would like to find some bounds on $\omega, \partial_x \omega, \partial_x^2 \omega, \partial_y \omega, \partial_y^2 \omega$. Well, | $\omega$ | is easy. But, is there anything I can say about the derivatives, in terms of $f$?

If one considers the case where $M$ is a square, then the usual fourier tricks are helpful. Is there something I can do for general regions $M$?

#### Solutions Collecting From Web of "Laplacian Boundary Value Problem"

As pointed out in comments by Shucao Cao and Ray Yang, Gilbarg-Trudinger is the place to look up such estimates: it does not have 3704 MathSciNet citations for nothing (as of today). In addition to global Hölder and Sobolev estimates (which require appropriate smoothness of $\partial M$ and $f$) there is a simple interior estimate for the Laplace equation, which applies in great generality. I quote it below.

Theorem 2.10. Let $u$ be harmonic in $\Omega\subseteq \mathbb R^n$ and let $\Omega’$ be a compact subset of $\Omega$. Then for any multi-index $\alpha$ we have
$$\sup_{\Omega’} |D^\alpha u|\le \left(\frac{n|\alpha|}{d}\right)^{|\alpha|}\sup_\Omega |u|$$ where $d=\operatorname{dist}(\Omega’,\partial\Omega)$.

By the maximum principle, you can replace $\sup_\Omega |u|$ with $\sup_{\partial \Omega}|f|$ where $f$ is the boundary data.