p-Laplacian

In mathematics, the p-Laplacian, or the p-Laplace operator, is a quasilinear elliptic partial differential operator of 2nd order. It is a nonlinear generalization of the Laplace operator, where ${\displaystyle p}$ is allowed to range over ${\displaystyle 1<p<\infty }$. It is written as

${\displaystyle \Delta _{p}u:=\nabla \cdot (|\nabla u|^{p-2}\nabla u).}$

Where the ${\displaystyle |\nabla u|^{p-2}}$ is defined as

${\displaystyle \quad |\nabla u|^{p-2}=\left[\textstyle \left({\frac {\partial u}{\partial x_{1}}}\right)^{2}+\cdots +\left({\frac {\partial u}{\partial x_{n}}}\right)^{2}\right]^{\frac {p-2}{2}}}$

In the special case when ${\displaystyle p=2}$, this operator reduces to the usual Laplacian.[1] In general solutions of equations involving the p-Laplacian do not have second order derivatives in classical sense, thus solutions to these equations have to be understood as weak solutions. For example, we say that a function u belonging to the Sobolev space ${\displaystyle W^{1,p}(\Omega )}$ is a weak solution of

${\displaystyle \Delta _{p}u=0{\mbox{ in }}\Omega }$

if for every test function ${\displaystyle \varphi \in C_{0}^{\infty }(\Omega )}$ we have

${\displaystyle \int _{\Omega }|\nabla u|^{p-2}\nabla u\cdot \nabla \varphi \,dx=0}$

where ${\displaystyle \cdot }$ denotes the standard scalar product.