I have a PDE class as a undergraduate student. The professor said that we want to approximate functions by using derivatives. (title: Estimates of derivatives of harmonic function)
The proof is as follows in my memory. So there can be some errors.
$$ \Delta u=0 \implies \Delta D_i u = 0$$
Let $B_R(y)\subset\subset\Omega$. Apply MVT to $D_iu$ :
$$ D_iu(y) = \frac 1 {w_nR^n}\int_{B_R(y)} D_i u \, dx$$
$D_iu = e_i \cdot Du = \operatorname{div}(e_i u)$
$(\because \operatorname{div}(\mathbf vf)=\mathrm{div} (\mathbf v) f + \mathbf v\cdot \nabla f)$
By the Divergence Theorem,
$$ D_iu(y) = \frac 1 {w_nR^n}\int_{B_R(y)} \mathrm{div}(e_iu) dx = \frac 1 {w_nR^n}\int_{\partial B_R(y)} e_iu \cdot \nu ds$$
So
$$|D_iu(y)| \leq \frac 1 {w_nR^n} \sup_{\partial B_R(y)}|u| \cdot \underbrace{ nw_nR^{n-1}}_{\text{surface area}} = \frac n R \sup_{\partial B_R(y)} |u| $$
At this time, I have a question. I don't know how the following inequality is derived, particularly in the second, even if $R < d(y,\partial\Omega)$.
$$ |D_u(y)| \leq \frac nR \sup_{\partial\Omega}|u| \leq \frac n {d(y,\partial\Omega)} \sup_\Omega |u| $$
A similar theorem appears on page 23 in Elliptic Partial Differential Eqns of Second Order 2001 by Gilbarg.
Best Answer
This
$$|D_iu(y)| \leq \frac nR \sup_{\partial\Omega}|u| \leq \frac n {d(y,\partial\Omega)} \sup_\Omega |u|\tag{$\ast$}$$
is wrong for $R < d(y,\partial\Omega)$ and harmonic $u \not\equiv 0$. The first of the two inequalities is okay if $u$ has decent boundary values on $\partial\Omega$ or we replace $\partial\Omega$ with $\partial B_R(y)$ or $\Omega$ there, but not the second. Since $u$ is harmonic, if it has nice (for example continuous) boundary values, we have
$$\sup_{\Omega} \lvert u\rvert = \sup_{\partial\Omega} \lvert u\rvert,$$
and since $R < d(y,\partial\Omega)$, the second inequality can only hold if the factor $\sup\limits_\Omega \lvert u\rvert$ is $0$. If we don't have good boundary values, the $\sup\limits_{\partial\Omega}$ would anyway have to be replaced with $\sup\limits_\Omega$ or $\sup\limits_{\partial B_R(y)}$. In either case, there are harmonic $u$ (constants, for example) for which the second inequality in $(\ast)$ doesn't hold.
However, the first inequality of $(\ast)$ [with the supremum taken over $\Omega$ to avoid demanding boundary regularity] holds for all $R < d(y,\partial\Omega)$, and hence
$$\lvert D_i u(y)\rvert \leqslant \inf_{R < d(y,\partial\Omega)} \frac{n}{R}\sup_{\Omega} \lvert u\rvert = \frac{n}{d(y,\partial\Omega)} \sup_{\Omega} \lvert u\rvert$$
gives us the estimate we are interested in.