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 $|\nabla \cdot |^{p-2}$ 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 $\varphi\in C^\infty_0(\Omega)$ 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) = \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 explicitely 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 $N$ an on $p$ only. Observe that for $p>2$ the solution is not twice differentiable in classical sense.

Notes

1 2 Evans, pp 356.

Sources

Evans, Lawrence C. (1982). "A New Proof of Local $C^{1,\alpha}$ Regularity for Solutions of Certain Degenerate Elliptic P.D.E.". Journal of Differential Equations 45: 356–373. doi:10.1016/0022-0396(82)90033-x. MR 672713.
Lewis, John L. (1977). "Capacitary functions in convex rings". Archive for Rational Mechanics and Analysis 66: 201–224. doi:10.1007/bf00250671. MR 0477094.

p-Laplacian

Energy formulation

Notes

Sources

Further reading