Overdetermined Hardy inequalities
type: Item
instance of: scholarly article
title: Overdetermined Hardy inequalities
zbMATH Open document ID: 0895.26004
published in: Journal of Mathematical Analysis and Applications
publication date: 1998-09-08T00:00:00Z
full work available at URL: http://hdl.handle.net/10338.dmlcz/126066
review text: The aim of the paper is to find necessary and sufficient conditions on the weights \(w\) and \(w_0\) for the validity of the higher-order Hardy inequality \[ \Biggl(\int^1_0| u|^qw_0\Biggr)^{1/q}\leq C\Biggl(\int^1_0| u^{(k+ 1)}|^p w\Biggr)^{1/p} \] on the class of all solutions of certain overdetermined boundary value problems. In Section 2 the case when \(k=0\) and \(u\) satisfies \(u'= f\) in \((0,1)\), \(u(0)= u(1)= 0\), is treated. The corresponding result is formulated as Theorem 2.1 and extends a result of P. Gurka (on the other hand, Theorem 2.1 is a particular case of Theorems 8.8, 8.17 and Remark 9.10 (ii) of \textit{B. Opic} and \textit{A. Kufner} [``Hardy-type inequalities'' (1990; Zbl 0698.26007)]). The core of the paper represents Section 3, where higher-order Hardy's inequality \((k\in\{1,2,\dots\})\) is investigated.
Mathematics Subject Classification ID: 26D10
zbMATH DE Number: 1115559
zbMATH Keywords: Hardy-type inequalities
zbMATH Keywords: weights
zbMATH Keywords: overdetermined boundary value problems
author: Alois Kufner
author: Gord Sinnamon
reviewed by: Bohumír Opic
MaRDI profile type: Publication
OpenAlex ID: W2728054787
cites work: Q5606769
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DOI: 10.1006/JMAA.1997.5551
description: scientific article; zbMATH DE number 1115559