# The $p$-Laplacian is strongly monotone

I am studying the solution of $p$-Laplacian by finding the minimizer of the following energy, among the space $W_0^{1,p}(\Omega)$, $p\geq 2$, $$E[u]=\frac{1}{p}\int_\Omega \lvert\nabla u\lvert^pdx+\int_\Omega fu\,dx$$ where $\Omega$ is open bounded nice boundary and $f\in W^{-1,p'}(\Omega)$. It is clear that $E[u]$ has a unique minimizer $\bar u$ by using the direct method. So far so good.

But next, my textbook states that for $p\geq 2$ the $p$-Laplacian is strongly monotone in the sense that $$\int_\Omega (\lvert \nabla \bar u\lvert^{p-2}\nabla \bar u-\lvert \nabla v\lvert^{p-2}\nabla v)(\nabla \bar u-\nabla v)dx\geq C\|\bar u-v\|_{W_{0}^{1,p}(\Omega)}^p \tag 1$$ for all $v\in W_0^{1,p}(\Omega)$

The book does not give any prove of $(1)$. It looks to me that this is somehow an variational inequality but I can not prove it either... Please help me.

Also, the book also states that by strongly monotone then the solution is unique. I can not see why monotone implies uniqueness of solution... (The way I see the solution is unique is that $E[u]$ is strictly convex..)

Any help is really welcome! Happy new year guys, by the way. :)

• You can find the answer of your question, in my answer here. – Tomás Jan 1 '15 at 3:15
• Oh i see. Thank you! – spatially Jan 1 '15 at 3:24
• @Tomás one more question. Why strong monotone could implies the uniqueness of solution? – spatially Jan 2 '15 at 14:28
• @Tomás I understand it. If both $u$ and $v$ are solutions then the left hand side is necessary be $0$ and hence the uniqueness follows! – spatially Jan 2 '15 at 15:27
• You can also prove uniqueness for the case where $p\in (1,2)$. Have you tried it? – Tomás Jan 2 '15 at 17:01