On p-harmonic functions in the plane and their stream functions
Suppose that D is a domain in \({\mathbb{R}}^ 2\) and \(p>1\). A function u is called p-harmonic in D if u is in the Sobolev space \(W^{1,p}_{loc}(D)\) and satisfies \(div(| \nabla u|^{p-2} \nabla u)=0\). The paper presents itself as a survey article but also contains new results. For instance it is proved that if \(\phi\) is p- harmonic in D but is not a polynomial of degree 1, then outside a set of isolated points, the function \(\phi\) is \({\mathbb{R}}\)-analytic and satisfies \(\phi_{xx}\phi_{yy}-\phi^ 2_{xy}\neq 0.\) Further p-harmonic functions of the form \(r^ kf(\theta)\), where \(x+iy=re^{i\theta},\) are studied in the case where \(p>2\), \(k>(p-2)/(p-1)\); these functions are used to illustrate the representation theorems of Stoilov type for such functions.
scientific article; zbMATH DE number 4077831
scientific article