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 $p$ is allowed to range over $1<p<\infty$ . It is written as

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

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

\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 $p=2$ , 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 $W^{1,p}(\Omega )$ is a weak solution of

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

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

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

where $\cdot$ denotes the standard scalar product.

Energy formulation

The weak solution of the p-Laplace equation with Dirichlet boundary conditions

{\begin{cases}-\Delta _{p}u=f&{\mbox{ in }}\Omega \\u=g&{\mbox{ on }}\partial \Omega \end{cases}}

in a domain $\Omega \subset \mathbb {R} ^{N}$ is the minimizer of the energy functional

J(u)={\frac {1}{p}}\,\int _{\Omega }|\nabla u|^{p}\,dx-\int _{\Omega }f\,u\,dx

among all functions in the Sobolev space $W^{1,p}(\Omega )$ satisfying the boundary conditions in the trace sense.^[1] In the particular case $f=1,g=0$ and $\Omega$ is a ball of radius 1, the weak solution of the problem above can be explicitly computed and is given by

u(x)=C\,\left(1-|x|^{\frac {p}{p-1}}\right)

where $C$ is a suitable constant depending on the dimension $N$ and on $p$ only. Observe that for $p>2$ the solution is not twice differentiable in classical sense.