The best constant of Sobolev inequality corresponding to a bending problem of a beam on an interval
type: Item
instance of: scholarly article
title: The best constant of Sobolev inequality corresponding to a bending problem of a beam on an interval
zbMATH Open document ID: 1209.34016
author: Yoshinori Kametaka
author: Atsushi Nagai
author: Kazuo Takemura
author: Kohtaro Watanabe
author: Hiroyuki Yamagishi
published in: Tsukuba Journal of Mathematics
publication date: 2010-05-27T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.tkbjm/1267209420
review text: The authors consider the boundary value problem \[ \text{BVP}(\alpha,\beta): u^{(4)}=f(x)\text{ for }x\in (-1,1) \] with boundary conditions \[ u^{(\alpha _{j})}(-1)=u^{(\beta _{j})}(1)=0\text{ for }j=0,1 \] associated to a bending problem of a beam on the interval \(I:=(-1,1)\), where \(f\) is a bounded continuous function on \(I\) and \((\alpha,\beta):=(\alpha_0,\alpha_1,\beta_0,\beta_1)\) takes one of the following ten values: Case I [\((0,1,0,1), (0,1,0,2), (0,1,1,3), (0,1,2,3), (0,2,0,2), (0,2,1,3)\)]; Case II [\((0,2,2,3),(1,3,1,3),(1,3,2,3)\)]; and Case III [\((2,3,2,3)\)]. They prove that BVP(\(\alpha,\beta\)) has a unique classical solution for the parameters in Case I. The parameters in Cases II and III have eigenfunctions \(\varphi\) corresponding to \(\lambda=0\) for the eigenvalue problem \(u^{(4)}(x)=\lambda u(x)\) with \(x\in I\), \(u^{(\alpha _{j})}(-1)=u^{(\beta _{j})}(1)=0,\) \(j=0,1.\) Then, if the conditions \(\int_{-1}^{1}f(y)\varphi(y)\,dy=0\) (solvability condition) and \(\int_{-1}^{1}u(y)\varphi(y)\,dy=0\) (orthogonality condition) are added to Cases II and III, BVP(\(\alpha,\beta\)) presents again a unique classical solution. In all cases, the solutions are given in terms of Green functions \(G(x,y)\) which are explicitly described. In the second part of the paper, it is proved that \(G(x,y)\) is the reproducing kernel for a suitable Hilbert space \(H\) and an inner product \((\cdot,\cdot)_H\). In particular, the authors show that \(G(y,y)=\int_{-1}^{1}|\partial^2_{x}G(x,y)|\,dx\) which allows them to prove that the Sobolev inequality \[ \left(\sup_{|y|\leq 1}|u(y)|\right)^2\leq C\int_{-1}^{1}|u''(x)|\,dx \] holds for any function \(u\) of \(H\) and a positive constant \(C\), independent of \(u\). Even more, they realize that the best constant \(C(\alpha,\beta)\) is the maximum of \(G(y,y),\) \(y\in [-1,1],\) and compute it in all cases.
Mathematics Subject Classification ID: 34B05
Mathematics Subject Classification ID: 34B27
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 41A44
zbMATH DE Number: 5712999
zbMATH Keywords: boundary value problem
zbMATH Keywords: bending of a beam
zbMATH Keywords: eigenvalue
zbMATH Keywords: eigenfunction
zbMATH Keywords: Green function
zbMATH Keywords: Hilbert space
zbMATH Keywords: inner product
zbMATH Keywords: reproducing kernel
zbMATH Keywords: Sobolev inequality
zbMATH Keywords: best constant
zbMATH Keywords: Schwarz inequality
zbMATH Keywords: symmetric orthogonalization method
reviewed by: Antonio Linero Bas
MaRDI profile type: Publication
OpenAlex ID: W1580353942
DOI: 10.21099/TKBJM/1267209420
description: scientific article; zbMATH DE number 5712999