Riemann zeta function and Lyapunov-type inequalities for certain higher order differential equations
type: Item
instance of: scholarly article
title: Riemann zeta function and Lyapunov-type inequalities for certain higher order differential equations
zbMATH Open document ID: 1254.34027
author: Hiroyuki Yamagishi
author: Yoshinori Kametaka
published in: Applied Mathematics and Computation
publication date: 2012-06-11T00:00:00Z
reviewed by: Antonio Cañada
Mathematics Subject Classification ID: 34B05
Mathematics Subject Classification ID: 34B27
zbMATH DE Number: 6045217
zbMATH Keywords: Lyapunov-type inequalities
zbMATH Keywords: Sobolev-type inequalities
zbMATH Keywords: higher order equations
zbMATH Keywords: Riemann zeta function
MaRDI profile type: Publication
full work available at URL: j.amc.2011.08.105
OpenAlex ID: W2095441254
cites work: Q4427597
cites work: Lyapunov-type integral inequalities for certain higher order differential equations
cites work: Q3087327
cites work: On a Lyapunov-type inequality for certain higher-order differential equations
DOI: 10.1016/J.AMC.2011.08.105
review text: The authors prove some Lyapunov-type inequalities for the equation NEWLINE\[NEWLINE u^{(2n)} (x) + r(x)u(x) = 0, \;a \leq x \leq b, NEWLINE\]NEWLINE where \(r(\cdot)\) is a continuous and nonnegative function in \([a,b]\) and for five types of boundary conditions: Dirichlet, periodic, anti-periodic, Neumann and Dirichlet-Neumann. The proof uses the best constant of some Sobolev-type inequalities for higher order derivatives where the Riemann zeta function plays an important role. More precisely, this best constant is associated with the study of some minimization problems for functionals of the form NEWLINE\[NEWLINE \frac{\int_{a}^{b} | u^{(n)} (x) | ^{2} \;dx} {\left ( \sup_{a \leq x \leq b} | u(x) | \right ) ^{2}} NEWLINE\]NEWLINE where the functions \(u(\cdot)\) belong to an appropriate Sobolev space with the corresponding boundary conditions.
description: scientific article; zbMATH DE number 6045217