# Local form of $p$-Laplacian operator in Riemannian manifold

Let $(M,g)$ be a connected oriented Riemannian manifold without boundary. The $p$-Laplacian of function $f:M\rightarrow\mathbb{R}$ is defined by $$\Delta_p f=\operatorname{div}\left(|\nabla f|^{p-2}\nabla f\right),$$ where $\nabla f$ is the gradient of $f$. I can not calculate the local form of $p$-Laplacian. I am trying to calculate it from the usual local form of laplacian operator but I am not getting any satisfactory form. Please help me.
Thank you.

• What do you mean by "local form"? – Anthony Carapetis Apr 5 '18 at 13:54
• The local form of laplacian of $f$ can be written as $g^{ij}\partial_j\partial_i f$. I want this kind of form for the $p$-Laplacian operator. – chandan mondal Apr 5 '18 at 14:08