I'm reading a paper concern with regularity of stable solutions to elliptic PDE. One of step in his proof make me confuse and highly suspect it is wrong. It's $$\frac{1}{|B_r|}\int_{B_r}|u–\bar{u}|dx\leq C\left(\int_{B_r}|Du|^2dx\right)^{\frac{1}{2}}\quad for~r\in(0,1)$$ where $\bar{u}$ is average in $B_r$ i.e. $$\frac{1}{|B_r|}\int_{B_r}udx$$ The author said just use Poincare and Holder inequality to obtain the conclusion, but I believe there will be a $r^{–1}$ cannot be control. Thanks for comments or solutions.
2026-03-26 12:46:21.1774529181
An equality derived by Poincare and Holder inequality
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