# Estimate on $C^1$-norm of solution of the Dirichlet problem for the Laplace equation

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.

• D. Gilbarg and N. Trudinger's book "Elliptic partial differential equations of second order" is one of the best references for such problems (see Chapter 3) Oct 19, 2021 at 11:13

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 $$x\mapsto A_\pm(x)=\phi(y)+v_\pm\cdot(x-y)$$, 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 Monge-Ampère 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.