Let $u \in H^{1}(\Omega)$ , $0<\tau <1$ and $B_{\tau R}$ the open ball of radius $\tau R$ under what conditions
(I) $||u||_{L^{p}(B_{\tau R})} = \sup\limits_{B_{R}} u$
and
(II) $||u||_{L^{p}(B_{\tau R})} \geq \sup\limits_{B_{R}} u$
?
Here R is a positive number.
My work:
I know that as $B_{\tau R}$ has finite measure, then if $p \to \infty$ $||u||_{L^{p}(B_{\tau R})} \to ||u||_{L^{\infty}(B_{\tau R})}$, but $ ||u||_{L^{\infty}(B_{\tau R})}\geq |u(x)|$ a.e.x in $B_{\tau R}$ from here , i don't know continue..