Smooth potentials with prescribed boundary behaviour
type: Item
instance of: scholarly article
title: Smooth potentials with prescribed boundary behaviour
zbMATH Open document ID: 1047.31005
DOI: 10.5565/PUBLMAT_
author: Anders Gustafsson
author: Stephen J. Gardiner
published in: Publicacions Matemàtiques
publication date: 2004-06-14T00:00:00Z
full work available at URL: https://eudml.org/doc/41499
review text: Let \(D \subset \mathbb R^n\) be a \(C^1\) domain where \(n \geq 2\). The main problem here is to study whether, given a continuous function \(g:\partial D \to (0, +\infty)\), there exists a function \(v \in C^1(\overline{ D})\) which is superharmonic on \(D\) and satisfies the boundary conditions \( v(z)= 0\) and \(\partial v/ \partial n_z = g(z) \) for \(z \in \partial D\), where \(\partial / \partial n_z \) denotes differentiation in the direction of the inward normal at \(z\). The authors prove that this is the case for so-called Lyapunov-Dini domains. Membership in the Lyapunov-Dini class requires that the angles between the directions \(n_z\) vary slowly enough (Dini condition). An example is given to show that the result is nearly sharp.
Mathematics Subject Classification ID: 31B15
zbMATH DE Number: 2074338
zbMATH Keywords: superharmonic function
zbMATH Keywords: Lyapunov-Dini domain
reviewed by: M. Vuorinen
MaRDI profile type: Publication
OpenAlex ID: W4214672224
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description: scientific article; zbMATH DE number 2074338