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<R\leq 1 $ 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?



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