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 |
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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 journal › Article
}
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 -