×

Inequalities for eigenvalue functionals. (English) Zbl 0944.34065

The author considers the system of eigenvalue problems \[ y'' + \lambda \rho_i(x) y = 0,\quad y(0) = 0 = y(\ell_i), \] subject to the constraints \(0 < \rho_i \leq H\) for \(i = 1,\dots,k\), \(\sum_i \int_0^{\ell_i}\rho_i(x)dx = M\). A sharp upper bound is proved for the quantity \(\Lambda := \min_{i}\lambda_1(\rho_i)\), where \(\lambda_1(\rho_i)\) denotes the lowest eigenvalue of the \(i\)th problem in the system.
The author also considers a system of problems of the form \((A_i^\beta(x)y'')'' + \lambda y'' = 0\) subject to \(y(0) = A_i^\beta(0)y''(0) = 0\) and \(y(\ell_i) = A_i^\beta(\ell_i)y''(\ell_i) = 0\), with the constraints \(\int_0^{\ell_i}A_i(x)dx = V>0\) and obtains a sharp lower bound on \(\sum_{i=1}^{k} \lambda_1(A_i)\) in terms of \(V\) and \(\beta\).

MSC:

34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
34B24 Sturm-Liouville theory
34L05 General spectral theory of ordinary differential operators
34L10 Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators
PDFBibTeX XMLCite
Full Text: DOI EuDML