On p-harmonic functions in the complex plane and curvature
Let u be a p-harmonic function, i.e. a weak solution of a p-harmonic equation \(div(| \nabla u|^{p-2}\nabla u)=0\), \(1<p<\infty\), in a plane domain. The author investigated some interesting p-harmonic analogues of the properties of harmonic functions \((p=2)\). For example let u be a p-harmonic function in \(G\subset {\mathbb{R}}^ 2\) and \(x_ 3=u(x_ 1,x_ 2)\) represents a p-harmonic surface in \({\mathbb{R}}^ 3.\) Theorem 5.3: The Gauss curvature \(K\leq 0\) for a p-harmonic surface. Either \(K=0\) at most at isolated points, or the surface is a plane. Theorem 5.5: If u is p-harmonic in the whole plane, then the total curvature of the corresponding p-harmonic surface is \(\iint K d\sigma \leq -2\pi,\) except when the surface is a plane.
scientific article; zbMATH DE number 4091843
scientific article