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 [math]\displaystyle{ p }[/math] is allowed to range over [math]\displaystyle{ 1 \lt p \lt \infty }[/math]. It is written as

[math]\displaystyle{ \Delta_p u:=\nabla \cdot (|\nabla u|^{p-2} \nabla u). }[/math]

Where the [math]\displaystyle{ |\nabla u|^{p-2} }[/math] is defined as

[math]\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} }[/math]

In the special case when [math]\displaystyle{ p=2 }[/math], 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 [math]\displaystyle{ W^{1,p}(\Omega) }[/math] is a weak solution of

[math]\displaystyle{ \Delta_p u=0 \mbox{ in } \Omega }[/math]

if for every test function [math]\displaystyle{ \varphi\in C^\infty_0(\Omega) }[/math] we have

[math]\displaystyle{ \int_\Omega |\nabla u|^{p-2} \nabla u\cdot \nabla\varphi\,dx=0 }[/math]

where [math]\displaystyle{ \cdot }[/math] denotes the standard scalar product.

Energy formulation

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

[math]\displaystyle{ \begin{cases} -\Delta_p u = f& \mbox{ in }\Omega\\ u=g & \mbox{ on }\partial\Omega \end{cases} }[/math]

in a domain [math]\displaystyle{ \Omega\subset\mathbb{R}^N }[/math] is the minimizer of the energy functional

[math]\displaystyle{ J(u) = \frac{1}{p}\,\int_\Omega |\nabla u|^p \,dx-\int_\Omega f\,u\,dx }[/math]

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

[math]\displaystyle{ u(x)=C\, \left(1-|x|^\frac{p}{p-1}\right) }[/math]

where [math]\displaystyle{ C }[/math] is a suitable constant depending on the dimension [math]\displaystyle{ N }[/math] and on [math]\displaystyle{ p }[/math] only. Observe that for [math]\displaystyle{ p\gt 2 }[/math] the solution is not twice differentiable in classical sense.

Sources

• Evans, Lawrence C. (1982). "A New Proof of Local [math]\displaystyle{ C^{1,\alpha} }[/math] 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. 
• Lewis, John L. (1977). "Capacitary functions in convex rings". Archive for Rational Mechanics and Analysis 66 (3): 201–224. doi:10.1007/bf00250671. Bibcode: 1977ArRMA..66..201L. 

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, ISBN 9780821886533, https://books.google.com/books?id=dolUcRSDPgkC .
• Uhlenbeck, K. (1977). "Regularity for a class of non-linear elliptic systems". Acta Mathematica 138: 219–240. doi:10.1007/bf02392316. 
• Notes on the p-Laplace equation by Peter Lindqvist
• Juan Manfredi, Strong comparison Principle for p-harmonic functions