# Some problems about energy estimates of elliptic equation

Recently I am reading a book of elliptic equations. In the beginning there is a famous Caccioppli inequality for weak solutions. The theorem is stated as follows

Suppose that $$u\in H^1(B(0,1))$$ satisfies $$\int_{B(0,1)}a_{ij}(x)D_iuD_j\varphi+\int_{B(0,1)}c(x)u\varphi dx=0,\forall \varphi\in C_c^{\infty}(B(0,1))$$ where $$a_{ij}(x)$$ is a matrix value function with uniformly elliptic condition, i.e. $$\sum_{i,j}a_{ij}(x)\xi_i\xi_j\geq\lambda|\xi|$$ for all $$x\in B(0,1),$$ $$\xi\in \mathbb{R}^d$$ and constant $$\lambda>0$$ and $$c(x)>0$$ is a bounded function. Then $$\int_{B_r}|\nabla u|^2dx\leq \frac{C}{(R-r)^2}\int_{B_R\backslash B_r}|u|^2dx$$ for all $$0\leq r with $$C$$ being a constant depending only on $$d$$.

By using the argument above, I have already know that we can get that $$\int_{B_r}|u|^2dx\leq C\left(\frac{R}{r}\right)^{\alpha}\int_{B_R}|u|^2dx$$ for some constant $$C$$ and $$\alpha>0$$. Physically thinking this implies that the increasing rates of the energy from $$B_r$$ to $$B_R$$. Generally speaking, we bound $$\int_{B_r}|u|^2$$ by $$\int_{B_R}|u|^2$$ for $$R>r$$ and estimate the increasing rates. I want to ask that can we have some reverse inequality? That is, can we give more assumptions to get that $$\int_{B_R}|u|^2$$ is bounded by $$\int _{B_r}|u|^2$$ in some sense when $$R>r$$? Can you give me some references or hints?