p-Laplacian

Summary

The $p$-Laplacian is the nonlinear operator $\Delta_p u = \operatorname{div}(\abs{\nabla u}^{p-2} \nabla u)$, for a real number $1 < p < \infty$. For $p = 2$ it is the ordinary Laplacian. It is the Euler–Lagrange operator of the energy $\int \abs{\nabla u}^p \, dx$, just as the Laplacian is for $\int \abs{\nabla u}^2 \, dx$. It is the model of a quasilinear elliptic operator that degenerates where $\nabla u = 0$ for $p > 2$ and becomes singular there for $p < 2$.