### 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 language | English |
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Pages (from-to) | 1279-1300 |

Number of pages | 22 |

Journal | SIAM Journal on Mathematical Analysis |

Volume | 29 |

Issue number | 5 |

Publication status | Published - Sep 1998 |

### Keywords

- Eigenvalue
- Isoperimetric inequalities
- Lagrange multiplier
- Rearrangement

### ASJC Scopus subject areas

- Mathematics(all)
- Analysis
- Applied Mathematics

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## Cite this

Karaa, S. (1998). Sharp estimates for the eigenvalues of some differential equations.

*SIAM Journal on Mathematical Analysis*,*29*(5), 1279-1300.