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

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

Where the ${\displaystyle |\nabla \cdot |^{p-2}}$ 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 $p=2$, 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 $\cdot$ denotes the standard scalar product.

Energy formulation

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

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

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

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

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

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

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. 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.

Further reading

• Ladyženskaja, O. A.; Solonnikov, V. A.; Ural'ceva, N. N. (1968), Linear and quasi-linear equations of parabolic type, Translations of Mathematical Monographs, 23, Providence, RI: American Mathematical Society, pp. XI+648, MR 0241821, Zbl 0174.15403.
• Uhlenbeck, K. (1977). "Regularity for a class of non-linear elliptic systems". Acta Mathematica. 138: 219–240. doi:10.1007/bf02392316. MR 0474389.
• Notes on the p-Laplace equation by Peter Lindqvist