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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 $?

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  • $\begingroup$ What type of solution you are looking for? Weak?Strong?Classical? $\endgroup$ – Shuhao Cao Jul 9 '13 at 4:00
  • $\begingroup$ Classical solution. $\endgroup$ – supersnooper Jul 9 '13 at 6:28
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    $\begingroup$ Schauder estimate in the PDE bible (Gilbarg-Trudinger) reads: $$\|\omega\|_{C^{2,\alpha}(M)}\leq C |f|_{C^{2,\alpha}(\partial M)}$$, where the first norm is the full norm for $C^{2,\alpha}$-functions (Holder continuous), second norm is the semi-norm on boundary. $\endgroup$ – Shuhao Cao Jul 9 '13 at 6:33
  • $\begingroup$ There are also Calderon-Zygmund type $W^{2,p}$ estimates if you want those (also in Gilbarg Trudinger, Chapter 9 in this case. Schauder estimates are in Chapter 4 for the Laplacian). $\endgroup$ – Ray Yang Jul 9 '13 at 10:51
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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.

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