Boundary conditions for a finite Dirichlet integral in the upper half plane
type: Item
instance of: scholarly article
title: Boundary conditions for a finite Dirichlet integral in the upper half plane
zbMATH Open document ID: 0704.30044
DOI: 10.1007/BF01246747
author: David Protas
author: Kwang-nan Chow
published in: Archiv der Mathematik
publication date: 1991-01-01T00:00:00Z
review text: Let k be an absolutely continuous function such that for some positive constant \(k_ 0\), \(k(t)\geq k_ 0\) for all real t. Suppose that \(k'\in L^ p(-\infty,\infty)\) for some \(p>1\). Suppose also that \(\lim_{t\to \infty}k(t)=\lim_{t\to -\infty}k(t)\) and that for some \(q>1\), \(k'(t)=O(| t|^{-q})\), \(| t| \to \infty\). It is shown there exists a function f, analytic in the upper half plane H with finite Dirichlet integral \[ \int_{H}\int | f'(z)|^ 2 dx dy, \] such that \(| f| =k\) almost everywhere on the real line. Examples are given which show that all the hypotheses are necessary. Also, differences between this result and what is known in the more familiar case of the unit disc are pointed out.
Mathematics Subject Classification ID: 30D50
Mathematics Subject Classification ID: 30D55
zbMATH DE Number: 4154837
zbMATH Keywords: outer function
zbMATH Keywords: boundary condition
reviewed by: Kwang-nan Chow
MaRDI profile type: Publication
cites work: A representation formula for the Dirichlet integral
cites work: The bounded, analytic, Dirichlet finite functions and their fibers
cites work: Q5618011
cites work: Q3922907
cites work: Q3874458
cites work: Q3933229
full work available at URL: https://doi.org/10.1007/bf01246747
OpenAlex ID: W1996701118
description: scientific article; zbMATH DE number 4154837