Representation of a p-harmonic function near a critical point in the plane
Let \(\Omega\) be a domain in \({\mathbb{R}}^ 2\) and let \(1<p<\infty\). A p- harmonic function \(\phi\) on \(\Omega\) is a weak solution of the equation \(div(| \nabla \phi |^{p-2}\nabla \phi)=0.\) Given such a function \(\phi\), there is an \(\alpha >0\) such that \(\phi\) belongs to \(C^{1,\alpha}\) on any compact subset of \(\Omega\). In fact, \(\phi\) is real analytic on \(\Omega\setminus E\), where E is the set of critical points of \(\phi\). It is the nature of \(\phi\) near critical points (which are known to be isolated) that is investigated in this paper. Using ``stream functions'' and the ``hodograph transformation'', the author establishes a representation theorem for \(\phi\) on a neighbourhood of a critical point \(z_ 0\). Among the consequences of this result is the determination of the exact Hölder exponent of \(\nabla \phi\) near \(z_ 0\) as a function of p and the ``order'' of the critical point.
scientific article; zbMATH DE number 4150788
scientific article