### 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 |
---|---|

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 |

### Fingerprint

### Keywords

- Eigenvalue
- Isoperimetric inequalities
- Lagrange multiplier
- Rearrangement

### ASJC Scopus subject areas

- Mathematics(all)
- Analysis
- Applied Mathematics

### Cite this

*SIAM Journal on Mathematical Analysis*,

*29*(5), 1279-1300.

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

Research output: Contribution to journal › Article

*SIAM Journal on Mathematical Analysis*, vol. 29, no. 5, pp. 1279-1300.

}

TY - JOUR

T1 - Sharp estimates for the eigenvalues of some differential equations

AU - Karaa, Samir

PY - 1998/9

Y1 - 1998/9

N2 - 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.

AB - 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.

KW - Eigenvalue

KW - Isoperimetric inequalities

KW - Lagrange multiplier

KW - Rearrangement

UR - http://www.scopus.com/inward/record.url?scp=0032330391&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0032330391&partnerID=8YFLogxK

M3 - Article

VL - 29

SP - 1279

EP - 1300

JO - SIAM Journal on Mathematical Analysis

JF - SIAM Journal on Mathematical Analysis

SN - 0036-1410

IS - 5

ER -