Generalization of maximum principle to other norms

Question

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).

Answer 1

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 $$ \Vert v \Vert_{W^{m, p}} \le C_2 \Vert \Delta \bar{g} \Vert_{W^{m - 2, p}} \le C_2 \Vert \bar{g} \Vert_{W^{m, p}} \le C_1 C_2 \Vert g \Vert_{W^{m - 1/p, p}}. $$