Where can I find this "standard elliptical argument"?

Question

I recently came across the following problem:

Assume $x_0 \in \mathbb{R}^N$ and \begin{equation} \Vert w \Vert_{L^\infty (B(x_0,1))} \leq C \end{equation} for some real constant $C>0$, where $B(x_0,1)$ denotes the unit ball around $x_0$.

My professor than stated that "standard elliptic theory" leads to \begin{equation} \vert \nabla w(x_0) \vert \leq K \left( \Vert \Delta w \Vert_{L^\infty (B(x_0,1))} + \Vert w \Vert_{L^\infty (B(x_0,1))} \right). \end{equation}

My question is

What kind of "standard elliptic" argument is this? Where can I find such an argument?

@ShuhaoCao Yes, I can assume that. Is there such an argument then? EDIT: oh sorry, I misread. No, it does not have compact support, but still may assumed to be smooth. — mjb, Jun 18 '13 at 13:19
I think your professor first used the Taylor expansion $$ w(x) = w(x_0) + \nabla w\cdot (x-x_0) + \frac{1}{2}(x-x_0)^T D^2w(\xi) (x-x_0) $$ If choosing $x = x_0 + \epsilon e_i$, i.e., the first order will have the $i$-th component of $\nabla u$: $$\epsilon|(\nabla u)i| = \left|w(x_0) - w(x) + \frac{1}{2}(x-x_0)^T D^2w(\xi) (x-x_0)\right|\ \leq \sup{x\in B(x_0,1)}|w| + \epsilon^2\sup_{x\in B(x_0,1)}|D^2 w|$$ However I am not sure how he can bound $|D^2 w|$ by Laplacian, for standard "elliptic argument" applies for $C^{2,\alpha}$ function. — Shuhao Cao, Jun 18 '13 at 14:24

score 3 · Accepted Answer · edited Apr 13 '17 at 12:20

The relevant estimates can be found in Section 3.4 (Gradient estimates for Poisson's equation) in the book Elliptic Partial Differential Equations of Second Order by Gilbarg and Trudinger. It is convenient to write $w=h+u$ where $h$ is harmonic with the same boundary values as $w$. By the maximum principle, $h$ is bounded by $\|w\|_{L^\infty}$, and the gradient estimate for harmonic functions (also found in GT, or see Interior gradient bound) gives an estimate for $|\nabla h(x_0)|$ in terms of $\sup|h|$.

To the function $u$ you can apply inequality $(3.16)$ in GT, which says $$\sup_\Omega d_x|\nabla u| \le C(\sup_\Omega |u|+\sup_\Omega d^2_x |\Delta u|)$$ where $d_x=\operatorname{dist}(x,\partial \Omega)$.

Where can I find this "standard elliptical argument"?

1 Answers1