The $p$-Laplacian, or the $p$-Laplace operator, is a quasilinear elliptic partial differential operator of second order. It is a nonlinear generalization of the Laplace operator, where $ p $ is allowed to range over $ 1 < p < \infty $.
The $p$-Laplacian, or the $p$-Laplace operator, is a quasilinear elliptic partial differential operator of second order. It is a nonlinear generalization of the Laplace operator, where $ p $ is allowed to range over $ 1 < p < \infty $. It is written as $$ \Delta _ p u := \nabla \cdot \left(|\nabla u|^{p-2}\nabla u \right) \text , $$ where the $ |\nabla u|^{p-2} $ is defined as $$ |\nabla u|^{p-2}=\left[\textstyle \left({\frac {\partial u}{\partial x_{1}}}\right)^{2}+\cdots +\left({\frac {\partial u}{\partial x_{n}}}\right)^{2}\right]^{\frac {p-2}{2}} \text . $$
In the special case when $ p = 2 $, this operator reduces to the usual Laplacian. 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 $$ \Delta _ p u=0 \text{ in } \Omega $$ 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.
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. 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.
Source: Wikipedia
