### Abstract

We consider the system of homogeneous Dirichlet boundary value problems (*) (FORMULA PRESENTED) in a smooth bounded domain W Í R^{N}, where L_{1} and L_{2} 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 R^{2}: (*) 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.

Research output: Contribution to journal › Article

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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 -