A proof of energy functional appearing in the regularity of elliptic and parabolic equations 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.
 A: I am not sure I understood your question correctly, but here are still some comments. Assume that 
$$ \tag 1
\int_{B(R)} |\nabla u|^p \, dx \le \int_{B(R)} |\nabla u + \nabla \phi|^p \, dx
$$
for all $\phi \in W_0^{1,p}(B(R))$, $1<p<\infty$. This means that $u$ is a minimizer of the $p$-Dirichlet energy and in particular a solution for the $p$-Laplace equation. The minimizing property (1) already implies that $u$ is locally bounded and $C^{1,\alpha}$.
Trivially $u=c$ satisfies (1) for any $c \in \mathbb{R}$. Now suppose further that
$$ \tag 2
\int_{B(r)} |\nabla u|^p \, dx= 0
$$ 
for some $0<r<R$. Clearly this means that $|\nabla u|=0$ in $B(r)$. 
Now, if your question is whether there are non-constant functions $u$ satisfying both (1) and (2), this is a relatively famous open problem if $p \neq 2$. If the dimension is $d=2$ or $p=2$, then it is known that $u$ must be a constant. Otherwise, very little is known to the best of my knowledge - even under the assumption that $u \in C^\infty$. For references regarding the known theory, try googling "unique continuation for $p$-Laplace equation". 
