A proof of energy functional appearing in the regularity of elliptic and parabolic equations

Question

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.

Answer 1

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".