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On certain p-harmonic functions in the plane. (English) Zbl 0661.31016

Lösungen \(\phi\) (x,y) der Gleichung \[ (| \nabla \phi |^{p- 2}\phi_ x)_ x+(| \nabla \phi |^{p-2}\phi_ y)_ y=0,\quad 1<p<\infty, \] werden p-harmonische Funktionen genannt. Wie im klassischen, linearen Fall \(p=2\), werden Funktionen \(\phi\) (x,y), \(\psi\) (x,y) bzw. p- und p’-harmonisch, \(1/p+1/p'=1,\) für welche \(| \nabla \phi |^{p-2}\phi_ x=\psi_ y\), \(| \nabla \phi |^{p- 2}\phi_ y=-\psi_ x\), konjugiert heißen. Eine besondere Klasse solcher Funktionenpaare wird vom Verf. betrachtet, welche die Paare \(Re(x+iy)^{\alpha}\), \(Im(x+iy)^{\alpha}\), \(\alpha >0\) des Falles \(p=2\) auf den Fall \(p\neq 2\) ausdehnen. Sie sind durch Benutzung von Polarkoordinaten in einem Sektor der xy-Ebene definiert. Die p- harmonischen Funktionen \(\phi\) (x,y) werden dann durch die Variablentransformation \(u=\phi_ x\), \(v=\psi_ y\), \(u+iv=ge^{i\theta}\) in neue Funktionen \(\phi\) (q,\(\theta)\) verwandelt, die mit den konjugierten \(\psi\) (q,\(\theta)\) der Gleichungen \(\psi_ q=q^{p-2}\phi_{\theta}\), \(\psi_{\theta}=(q^{p-1}/(1-p))\phi_ q\) verbunden sind. Bei Variablentrennung erhält man dann eine Folge von speziellen Lösungen, mit deren Hilfe das Paar \(\phi\), \(\psi\) durch Reihenentwicklungen dargestellt werden kann.
Reviewer: G.Cimmino

MSC:

31C05 Harmonic, subharmonic, superharmonic functions on other spaces
31A30 Biharmonic, polyharmonic functions and equations, Poisson’s equation in two dimensions
35J60 Nonlinear elliptic equations
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References:

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