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 generalization of the Laplace operator, where p is allowed to range over $1 < p < \infty$. It is written as

$\nabla \cdot (|\nabla u|^{p-2} \nabla u).$

In the special case when p = 2, it is the regular Laplacian. ^[1]

Energy formulation

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

$\nabla \cdot (|\nabla u|^{p-2} \nabla u) = 0$

in a domain Ω is the minimizer of the energy functional

$J(u) = \int |\nabla u|^p \,dx$

among all functions in the Sobolev space W^1,p(Ω) satisfying the boundary conditions in the trace sense.^[1]

Notes

1. ^ ^{a b} Evans, pp 356.

Sources

• Evans, Lawrence C. (1982). "A New Proof of Local C^1,α Regularity for Solutions of Certain Degenerate Elliptic P.D.E.". Journal of Differential Equations 45: 356–373. 
• Lewis, John L. (1977). "Capacitary functions in convex rings". Archive for Rational Mechanics and Analysis 66: 201–224. 

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