How do I derive p-Laplacian?

Question

How do I obtain the p-Laplacian equation $$\Delta_p u = \nabla \cdot (|\nabla u|^{p-2} \nabla u) = 0$$ as the minimiser of the integral $$\int_{\Omega}|\nabla u|^p$$ ?

I can't expand $|\nabla u + t\nabla v|^p$ nicely.

Answer 1

Use the fact that $\nabla |x|^p = p|x|^{p-2}x$. This implies $$\frac{d}{dt} |\nabla u + t \nabla v|^p = p|\nabla u + t \nabla v|^{p-2}(\nabla u + t \nabla v) \cdot \nabla v$$ and consequently $$\left.\frac{d}{dt} |\nabla u + t \nabla v|^p\right|_{t = 0} = p |\nabla u|^{p-2} \nabla u \cdot \nabla v = 0.$$ Since $$\int_\Omega|\nabla u|^{p-2} \nabla u \cdot \nabla v \, dx = 0$$ you can integrate by parts to get $$ \int_\Omega \nabla \cdot( |\nabla u|^{p-2} \nabla u) v \,dx = 0$$ for all smooth compactly supported $v$. Thus $$\nabla \cdot( |\nabla u|^{p-2} \nabla u) = 0$$ in $\Omega$.