p-harmonic function

Summary

A $p$-harmonic function is a solution of the p-Laplace equation $\operatorname{div}(\abs{\nabla u}^{p-2} \nabla u) = 0$, where $1 < p < \infty$. For $p = 2$ these are the classical harmonic functions. They are the minimizers of the energy $\int \abs{\nabla u}^p \, dx$ with given boundary values, and they are the central objects of nonlinear potential theory. The equation is nonlinear for $p \ne 2$, so sums of $p$-harmonic functions are in general not $p$-harmonic.