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 . It is written as

:$abla\; cdot\; (|\; abla\; u|^\{p-2\}\; abla\; u)$

In the special case when $p=2$, it is the regular Laplacian.

Energy formulation

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

:$abla\; cdot\; (|\; abla\; u|^\{p-2\}\; abla\; u)\; =\; 0$

in a domain $Omega$ is the minimizer of the energy functional

:$J(u)\; =\; int\; |\; abla\; u|^p\; ,dx$

among all functions in the Sobolev space $W^\{1,p\}(Omega)$ with the appropriate boundary values.

Sources

*cite journal | last = Evans | first = Lawrence C. | authorlink = Lawrence C. Evans | title = A New Proof of Local $C^\{1,alpha\}$ Regularity for Solutions of Certain Degenerate Elliptic P.D.E. | journal = Journal of Differential Equations | volume = 45 | pages = 356-373 | date = 1982
*cite journal | last = Lewis | first= John L. | title = Capacitary functions in convex rings | journal = Archive for Rational Mechanics and Analysis | volume = 66 | pages = 201-224 | date = 1977