Abstract
We consider the system of homogeneous Dirichlet boundary value problems (*) (FORMULA PRESENTED) in a smooth bounded domain W Í RN, where L1 and L2 are formally selfadjoint second-order strongly uniformly elliptic operators. Using linear perturbation theory, continuation methods, and the Courant-Hilbert variational eigenvalue characterization, we give a detailed qualitative and quantitative description of the real generalized spectrum of (*), i.e., the set ((l, m) Î R2: (*) has a nontrivial solution). The generalized spectrum, a term introduced by Protter in 1979, is of considerable interest in the theory of linear partial differential equations and also in bifurcation theory, as it is the set of potential bifurcation points for associated semilinear systems.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 345-363 |
| Number of pages | 19 |
| Journal | Transactions of the American Mathematical Society |
| Volume | 303 |
| Issue number | 1 |
| DOIs | |
| State | Published - 1987 |
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ASJC Scopus subject areas
- Mathematics(all)
- Applied Mathematics
Cite this
On the generalized spectrum for second-order elliptic systems. / Cantrell, Robert; Cosner, George.
In: Transactions of the American Mathematical Society, Vol. 303, No. 1, 1987, p. 345-363.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - On the generalized spectrum for second-order elliptic systems
AU - Cantrell, Robert
AU - Cosner, George
PY - 1987
Y1 - 1987
N2 - We consider the system of homogeneous Dirichlet boundary value problems (*) (FORMULA PRESENTED) in a smooth bounded domain W Í RN, where L1 and L2 are formally selfadjoint second-order strongly uniformly elliptic operators. Using linear perturbation theory, continuation methods, and the Courant-Hilbert variational eigenvalue characterization, we give a detailed qualitative and quantitative description of the real generalized spectrum of (*), i.e., the set ((l, m) Î R2: (*) has a nontrivial solution). The generalized spectrum, a term introduced by Protter in 1979, is of considerable interest in the theory of linear partial differential equations and also in bifurcation theory, as it is the set of potential bifurcation points for associated semilinear systems.
AB - We consider the system of homogeneous Dirichlet boundary value problems (*) (FORMULA PRESENTED) in a smooth bounded domain W Í RN, where L1 and L2 are formally selfadjoint second-order strongly uniformly elliptic operators. Using linear perturbation theory, continuation methods, and the Courant-Hilbert variational eigenvalue characterization, we give a detailed qualitative and quantitative description of the real generalized spectrum of (*), i.e., the set ((l, m) Î R2: (*) has a nontrivial solution). The generalized spectrum, a term introduced by Protter in 1979, is of considerable interest in the theory of linear partial differential equations and also in bifurcation theory, as it is the set of potential bifurcation points for associated semilinear systems.
UR - http://www.scopus.com/inward/record.url?scp=84967734504&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84967734504&partnerID=8YFLogxK
U2 - 10.1090/S0002-9947-1987-0896026-2
DO - 10.1090/S0002-9947-1987-0896026-2
M3 - Article
AN - SCOPUS:84967734504
VL - 303
SP - 345
EP - 363
JO - Transactions of the American Mathematical Society
JF - Transactions of the American Mathematical Society
SN - 0002-9947
IS - 1
ER -