Let $\Omega\subset \mathbb{R}^n$ be a bounded domain with $C^\infty$smooth boundary. Let $\phi\in C^\infty(\partial \Omega)$. Let $u$ be the solution of the Dirichlet problem of the Laplace equation \begin{eqnarray} \Delta u=0 \mbox{ in }\Omega,\\ u=\phi \mbox{ in } \partial \Omega. \end{eqnarray} I am looking for an upper estimate on $\u\_{C^1(\bar\Omega)}$ in terms of $\phi$ (the dependence on the geometry of $\Omega$ is not important for me for the moment). A reference will be very helpful.
1 Answer
The first derivatives of $u$ are harmonic too. Therefore the Maximum Principle tells us $$\\nabla u\_{C^0(\bar\Omega)}=\left\\left.\nabla u\right_{\partial\Omega}\right\_{C^0(\partial\Omega)}.$$ It is thus enough to estimate the restriction of $\nabla u$ along the boundary. Of course, tangential derivatives of $u$ equal tangential derivatives of $\phi$. Thus there remains to estimate the normal derivative $\nabla u\cdot N$.
There is a nice trick when $\Omega$ is a uniformly convex domain. By this, I mean that there exists a finite $R$ such that, for every boundary point $y$, $\Omega$ is contained in a ball $B(a(y);R)$ of radius $R$, whose boundary $S(a(y);R)$ passes through $y$ (in other words, $a(y)y=R$). Under this condition, you can find two affine functions $x\mapsto A_\pm(x)=\phi(y)+v_\pm\cdot(xy)$, with $v_\pm$ bounded in terms of $R$ and of $\\phi\_{C^2}$, and such that $A_(x)\le\phi(x)\le A_+(x)$ along the boundary. Of course, $v_+v_$ is a normal vector. By the MP, one derives $A_(x)\le u(x)\le A_+(x)$ in the interior, which gives the desired estimate of the normal derivative: $$v_+\cdot N\le\nabla u\cdot N\le v_\cdot N.$$ The final result is thus $$\sup_{x\in\Omega}\nabla u(x)\le C\left(\sup_{x\in\partial\Omega}\nabla \phi(x)+\frac1R\sup_{x\in\partial\Omega}\nabla^2\phi(x)\right)$$ for some absolute constant $C$.
Edit. The strength of this argument is that it involves only the fact that $u$ and $\nabla u$ obey to the MP. Thus it applies to every fully nonlinear elliptic equation $$F(\nabla^2u)=0.$$ Here, elliptic means than $S\mapsto F(S)$ is monotonous (strictly) increasing on its domain $D\subset{\bf Sym}_n$. For instance, it applies to the MongeAmpère equation ($F(S)=\det S$), where $D={\bf SPD}_n$ and the convexity of $\Omega$ is a necessary condition of solvability for arbitrary Dirichlet boundary condition. Importantly, the constant in the estimate given above does not depend upon the specific PDE.

$\begingroup$ Thank you, this is nice. In my situation however the domain does not have to be convex. $\endgroup$– asvOct 19, 2021 at 13:22