Consider the Laplace equation $\Delta u=0$ in $\Omega \subset \mathbb{R}^d$ with Dirichlet boundary conditions, i.e. $u=g$ on $\delta \Omega$. By the maximum principle we know that the solution $u$ satisfies $$\|u\|\_{L^{\infty}(\Omega)}\leq\|g\|\_{L^{\infty}(\delta\Omega)}.$$
My question is if there exist similar estimates for Sobolev norms, i.e. if we have $$\|u\|\_{W^{m,p}(\Omega)}\leq C(m,p,\Omega)\|g\|\_{W^{m,p}(\delta\Omega)},$$
where $C(m,p,\Omega)$ is a constant depending only on $m,p$ and $\Omega$. 


I am trying to numerical solve Laplace equation (with manifold-valued data) and my numerical solution can not fit the boundary data exactly. Therefore I have to approximate it and I was wondering how this affects the solution in the interior. Unfortunately I couldn't find any solution for this problem in the literature (even for the real-valued case).



Generalization of maximum principle to other norms
One way to reformulate this is to consider a good extension $\bar{g}$ to the whole $\Omega$ of the function $g$, and then set $v = u - \bar{g}$.


The function $v$ solves then the problem
$$
\left\{
\begin{aligned}
-\Delta v &= \Delta \bar{g} && \text{in $\Omega$},\\
v & = 0 && \text{on $\partial \Omega$},
\end{aligned}
\right.
$$
You can then estimate $v$ in terms of $\Delta \bar{g}$ and thus $u$ in terms of $\bar{g}$.


If you are interested in Sobolev spaces, the construction of $g$ would rely on *trace theory*. For example, if your boundary is smooth enough and $g$ is in the fractional Sobolev space $W^{m - \frac{1}{p}, p} (\partial \Omega)$, then you can find $\bar{g} \in W^{m, p} (\Omega)$ with 
$$
 \Vert \bar{g}\Vert\_{W^{m, p}} \le C\_1 \Vert g \Vert\_{W^{m - 1/p, p}}.
$$


You can then apply the classical regularity theory that goes back to Calderón and Zygmund (see for example the book by Gilbarg and Trudinger) to obtain that $$
 \Vert v \Vert\_{W^{m, p}} \le C\_2 \Vert \Delta \bar{g} \Vert\_{W^{m - 2, p}}
\le C\_2 \Vert \bar{g} \Vert\_{W^{m, p}}
\le C\_1 C\_2 \Vert g \Vert\_{W^{m - 1/p, p}}.
$$


