After the proof of the Green's identity in the book "Han Q., Lin F. – Elliptic partial differential equations – AMS (1997)", they state at page 9:
We may employ the local version of the Green's identity to
get gradient estimates without using mean value property.Suppose $u \in C(\bar{B_1})$ harmonic in $B_1$. For any fixed
radius $0<r<R<1$ choose a cut-off function $\varphi \in
C_{0}^{\infty}(B_R)$ such that $\varphi =1$ in $B_r$ and $0 \leq
\varphi \leq 1$.Apply Green's formula to $u$ and $\varphi \Gamma(a, \cdot)$ in $B_1 \setminus
B_{\rho}(a)$ for $a \in B_r$ and $\rho$ small enough. We proceed as in
the proof of theorem 1.17 ( which is the proof of Green's identity) and we obtain\begin{align} u(a)=- \int_{r < |x| < R} u(x) \Delta_x \big(\varphi(x)
\Gamma(a,x)\big)dx \quad \quad (\star) \end{align} for any $a \in B_r(0)$Hence one may prove $\sup_{B_{1/2}}{Du} \leq C \max_{B_1} {|u|}$
where $\Gamma(a, x)$ is the fundamental solution
I can't understand how to derive that bound!
I think I should take the derivative of $u(a)$ w.r.t $a_i$ and get
\begin{align} | \partial_{a_i} u(a)| \leq \int_{r<|x|<R} |u(x)|\left |\Delta_x\big(\varphi(x) \partial_{a_i} \Gamma(a,x)\big)\right| dx
\end{align}
Now I should take outside the maximum of $|u|$ over the closure of the unitary ball, but I don't know how to treat the Laplacian term in the right way
Best Answer
You barely have to treat it. You have already shown $$|\partial_{a_i} u(a)| \le \int_{r < |x| < R} |u(x)| \left| \Delta_x(\varphi(x)\partial_{a_i} \Gamma(a,x))\right|dx,$$ so $$\sup_{a \in B_{1/2}} |\partial_{a_i} u(a)| \le \sup_{a \in B_{1/2}}\int_{r < |x| < R} |u(x)| \left| \Delta_x(\varphi(x)\partial_{a_i} \Gamma(a,x))\right|dx.$$ Let $r=\frac{3}{4}$ and $R = \frac{4}{5}$. $\varphi$ is fixed and so all you need is that $$\sup_{a \in B_{1/2}} \int_{3/4 < |x| < 4/5} |\Delta_x(\varphi(x)\partial a_i \Gamma(a,x))|dx < \infty$$ (then just call it $C$), but the integrand changes continuously in $a$ and $\overline{B_{1/2}}$ is compact, so you're good.