Lyapunov type inequality for the equation including 1-dim \(p\)-Laplacian
type: Item
instance of: scholarly article
zbMATH Open document ID: 1258.34044
author: Kohtaro Watanabe
publication date: 2012-08-31T00:00:00Z
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 34A40
Mathematics Subject Classification ID: 34B27
zbMATH DE Number: 6074814
zbMATH Keywords: \(p\)-Laplacian
zbMATH Keywords: Lyapunov-type inequality
zbMATH Keywords: Sobolev inequality
zbMATH Keywords: best constant
MaRDI profile type: Publication
full work available at URL: mia-15-58
OpenAlex ID: W2322502895
DOI: 10.7153/MIA-15-58
title: Lyapunov type inequality for the equation including 1-dim \(p\)-Laplacian
published in: Mathematical Inequalities \& Applications
review text: The authors obtain a Lyapunov-type inequality for equations of the form NEWLINE\[NEWLINE(-1)^m(|u^{(m)}(x)|^{p-2}u^{(m)}(x))^{(m)}=r(x)|u(x)|^{p-2}u(x),\quad a\leq x\leq b,\tag{\(*\)} NEWLINE\]NEWLINE under clamped boundary condition \(u^{(i)}(a)=u^{(i)}(b)=0\), \(i = 0,1,2,\dots,m-1\). The main result in the paper is stated as follows:NEWLINENEWLINETheorem. Let \(m = 1,2, 3\), \(p > 1\), \(q\) be a conjugate exponent of \(p\), that is, \(\frac1p+\frac1q=1\). If \(u\in W_0^{m,p}(a,b)\) is a non-trivial solution of \((*)\), then NEWLINE\[NEWLINE\frac{1}{C(m,p)^p}<\int_a^br(x)dx,NEWLINE\]NEWLINE where \(C(m,p)\) is the best constant of the \(L^p\)-Sobolev inequality NEWLINE\[NEWLINE(\sup_{a\leq x\leq b}|v(x)|)\leq C \biggl(\int_a^b|v^{(m)}(x)|^pdx\biggr)^{\frac1p}, \quad v\in W_0^{m,p}(a,b).NEWLINE\]NEWLINE \(W^{m,p}(a,b)\) is the Sobolev space defined on the interval \([a, b]\), that is, \(W^{m,p}(a,b)=\{u\mid u^{(i)}\in L^p(a,b), i=0,1,\dots,m\}\) and \(W_0^{m,p}(a,b)\) is a sub-space of \(W^{m,p}(a,b)\) whose derivatives up to \(m-1\) vanish at \(x = a, b\), where \(u^{(i)}\) denotes \(i\)-th derivative of \(u\) in the sense of distributions.
description: scientific article; zbMATH DE number 6074814