An interior estimate for the gradient of weak solutions to elliptic equations Let $L$ be an elliptic operator in divergence form i.e. $$Lu = -\sum_{i,j=1}^n D_i(a^{ij}(x)D_ju) + \sum_{i=1}^nb^i(x)D_iu + c(x)u$$ with $a^{ij}, b^i, c \in  C^1(Ω) ∩ L^
∞(Ω)$ and $L$ uniformly elliptic. Let $f \in L^2(\Omega)$ and suppose $u\in H^1(\Omega)$ solves $Lu=f$ weakly.
I am asked to prove for $U \subset\subset V \subset\subset \Omega$  the following estimate:
$$\int_U|\nabla u|^2\leq C\int_V|u|^2 + C\int_Vf^2$$
As a hint I am asked to choose in the weak formulation $v = \eta^2u$ for $\eta$ an appropriately chosen cut-off function.
I assume my cutoff $\eta$ will have to be a smooth function taking values in $[0,1]$ that is 1 on $U$ and 0 outside $V$
How do I proceed?
Any help would be appreciated.
 A: This is usually referred to as the Caccioppoli estimate. As recommended test against $v=\eta^2 u$ with $\eta$ a smooth cutoff function which is identically $1$ in $U$ and $0$ outside $V$. Then, we have for the left hand side of $Lu=f$ (using the Einstein convention of summation over repeated indices)
\begin{align}
\int_{\Omega} \eta^2 u Lu \, \mathrm{d}x = & \int_{\Omega}\eta^2 D_i u \, a^{ij}(x)D_j u \, \mathrm{d}x + 2\int_{\Omega}(\eta D_i\eta) \, u a^{ij}(x)D_j u \, \mathrm{d}x + \int_{\Omega}\eta^2 u b^i(x)D_i u \, \mathrm{d}x \\
&+ \int_{\Omega}\eta^2 c(x) u^2 \, \mathrm{d}x \\
\geq& \lambda \int_{\Omega}\eta^2|\nabla u|^2 \, \mathrm{d}x + 2\int_{\Omega}(\eta D_i\eta) \, u a^{ij}(x)D_j u \, \mathrm{d}x + \int_{\Omega}\eta^2 u b^i(x)D_i u \, \mathrm{d}x \\
&+ \int_{\Omega}\eta^2 c(x) u^2 \, \mathrm{d}x \, ,
\end{align}
for some $\lambda>0$, where we used the fact that $a$ is coercive ($L$ is uniformly elliptic). Transferring the remaining terms to the right hand side we have
\begin{align}
\lambda \int_{\Omega}\eta^2|\nabla u|^2 \, \mathrm{d}x \leq &  -2\int_{\Omega}(\eta D_i\eta) \, u a^{ij}(x)D_j u \, \mathrm{d}x - \int_{\Omega}\eta^2 u b^i(x)D_i u \, \mathrm{d}x \\
&- \int_{\Omega}\eta^2 c(x) u^2 \, \mathrm{d}x + \int_{\Omega}\eta^2 u f \mathrm{d}x  \\
\leq &C \left(\int_{\Omega}\eta|\nabla \eta||u||\nabla u| \, \mathrm{d}x + \int_{V}u^2 \, \mathrm{d}x + \int_{V} uf \mathrm{d}x  \right) \, ,
\end{align}
where we have used both the bounds of $a,b,c$ and the fact that $\eta$ and its derivatives are supported inside $V$. Finally, we can apply Young's inequality to the first and last term on the right hand side to obtain the desired estimate.  Note that you have to absorb the gradient term that comes from applying Young's inequality to the first term into the left hand side and use the fact that $\eta$ is $1$ on $U$.
