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


 * $$\Delta_p u:=\nabla \cdot (|\nabla u|^{p-2} \nabla u).$$

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. 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 $$\varphi\in C^\infty_0(\Omega)$$ 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. 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.