This article is cited in 1 paper

Mathematics

On a minimization problem for a functional generated by the Sturm–Liouville problem with integral condition on the potential

S. S. Ezhak

Moscow State University of Economics, Statistics and Informatics, 7, Nezhinskaya Street, Moscow, 119501, Russian Federation

Abstract: In this article we consider the minimization problem of the functional $R[Q,y]=\frac{\int_{0}^{1}y'^2dx- \int_{0}^{1}Q(x)y^2dx}{\int_{0}^{1}y^2dx}$ generated by a Sturm–Liouville problem with Dirichlet boundary conditions and with an integral condition on the potential. Estimation of the infimum of functional in some class of functions $y$ and $Q(x)$ is reduced to estimation of a nonlinear functional non depending on the potential $Q(x)$. This leads to related parameterized nonlinear boundary value problem. Upper and lower estimates for $\inf_{y\in H_{0}^{1}(0,1)}R[Q,y]$ are obtained for different values of parameter.

Keywords: variational problem, minimization of a functional, problem of Sturm–Liouville, extremal estimates, infimum, spectral theory.

UDC: 517.5

Received: 16.07.2015

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