I have got trapped in this problem for nearly two years when I dealt with regularity of solutions of elliptic and parabolic equations. I have not found a nice proof to support this assertion. Now I am looking for someone who could help me to solve this problem.

**Prolem:** Let $B_{R}$ be a ball in the Euclid space **$R$**$^{n}$ with radius $R$. Let $u-v \in W^{1,p}_{0}(B_{R})$ with $0\leq
\|v\|_{\infty}\leq M$. Assume
\begin{eqnarray}\int_{B_{R}}|\nabla u|^{p}dx\leq \int_{B_{R}}|\nabla u+\nabla
\phi|^{p}dx
\end{eqnarray}
for any $\phi\in C^{\infty}_{0}(B_{R})$ . Whether does the
following equality hold
\begin{eqnarray}\int_{B_{r}}|\nabla u|^{p}dx=0 \end{eqnarray} for any
$r<R$ (or some $r$)?

If this assertion is true, how to prove? If not, is there any counterexample?

Int: One may make more (regularity) assumptions on $u$, such as $\|u\|_{1,\alpha}\leq C,\ \|u\|_{\infty}\leq C, \ u\in C^{\infty}$, etc..

Thank you very much for your consideration.