Sharp estimates for the eigenvalues of some differential equations

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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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Keywords

  • Eigenvalue
  • Isoperimetric inequalities
  • Lagrange multiplier
  • Rearrangement

ASJC Scopus subject areas

  • Mathematics(all)
  • Analysis
  • Applied Mathematics

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