Sharp estimates for the eigenvalues of some differential equations

Research output: Contribution to journalArticle

23 Citations (Scopus)

Abstract

We present optimal upper and lower bounds for the eigenvalues of the differential equations y″ - q(x)y + λρ(x)y = 0 and (q(x)y′)′ + λρ(x)y = 0 on a finite interval with Dirichlet boundary conditions when the coefficient functions q(x) and ρ(x) are nonnegative and are subjected to some kind of additional constraints. One of the basic ideas used in our work consists in reducing the problem of maximizing λ(q, ρ) to an elementary problem of calculus of variations. This allows us to establish sufficient optimality conditions for our problems. We establish in the last part of this paper some comparison results for eigenvalues via symmetrization.

Original languageEnglish
Pages (from-to)1279-1300
Number of pages22
JournalSIAM Journal on Mathematical Analysis
Volume29
Issue number5
Publication statusPublished - Sep 1998

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Differential equations
Boundary conditions
Differential equation
Eigenvalue
Estimate
Sufficient Optimality Conditions
Q-function
Symmetrization
Comparison Result
Calculus of variations
Dirichlet Boundary Conditions
Upper and Lower Bounds
Non-negative
Interval
Coefficient

Keywords

  • Eigenvalue
  • Isoperimetric inequalities
  • Lagrange multiplier
  • Rearrangement

ASJC Scopus subject areas

  • Mathematics(all)
  • Analysis
  • Applied Mathematics

Cite this

Sharp estimates for the eigenvalues of some differential equations. / Karaa, Samir.

In: SIAM Journal on Mathematical Analysis, Vol. 29, No. 5, 09.1998, p. 1279-1300.

Research output: Contribution to journalArticle

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