Let $\Omega\subset \mathbb{R}^d$ be open and bounded with $C^\infty$ boundary $\partial\Omega$, $\phi\colon \partial\Omega \rightarrow \mathbb{R}$ continuous and $u^\phi$ the solution to Laplace's equation $\Delta u^\phi=0$ on $\Omega$ with Dirichlet boundary conditions $u^\phi|_{\partial\Omega}=\phi$. Is it true that there exists $C>0$ independent of $\phi$ such that $$ |u^\phi|_{H^1}:=\|\nabla u^\phi\|_{L^2}\leq C\|\phi\|_{L^\infty}? $$ More generally if $M\subset \mathbb{R}^n$ is a Riemannian submanifold of $\mathbb{R}^n$ and $$ u^\phi:=\mathop{argmin}_{\substack{v \in H^1(\Omega,M)\\v|_{\partial\Omega}=\phi} } |v|_{H^1}, $$ is it true that $$ |u^{\phi_1}-u^{\phi_2}|_{H^1}\leq C\|\phi_1-\phi_2\|_{L^\infty}? $$ I am trying to derive a discretization error estimate for a numerical scheme. Since the boundary condition can in general not be implemented exactly I would like to estimate the error due to errors in the boundary data.

Let $\Omega\subset \mathbb{R}^d$ be open and bounded with $C^\infty$ boundary $\partial\Omega$, $\phi\colon \partial\Omega \rightarrow \mathbb{R}$ continuous and $u^\phi$ the solution to Laplace's equation $\Delta u^\phi=0$ on $\Omega$ with Dirichlet boundary conditions $u^\phi|_{\partial\Omega}=\phi$. Is it true that there exists $C>0$ independent of $\phi$ such that $$ |u^\phi|_{H^1(\Omega)}:=\|\nabla u^\phi\|_{L^2(\Omega)}\leq C\|\phi\|_{L^\infty(\partial\Omega)}? $$ More generally, if $M\subset \mathbb{R}^n$ is a Riemannian submanifold of $\mathbb{R}^n$ and $$ u^\phi:=\mathop{argmin}_{\substack{v \in H^1(\Omega,M)\\v|_{\partial\Omega}=\phi} } |v|_{H^1(\Omega,\mathbb{R}^n)}, $$ is it true that $$ |u^{\phi_1}-u^{\phi_2}|_{H^1(\Omega,\mathbb{R}^n)}\leq C\|\phi_1-\phi_2\|_{L^\infty(\partial\Omega,\mathbb{R}^n)}? $$ I am trying to derive a discretization error estimate for a numerical scheme. Since the boundary condition can in general not be implemented exactly I would like to estimate the error due to errors in the boundary data.