On the Generalized Spectrum for Second-Order Elliptic Systems
Robert Stephen Cantrell, Chris Cosner · Transactions of the American Mathematical Society · 1987
We consider the system of homogeneous Dirichlet boundary value problems $({\ast })$ \[ {L_1}u = \lambda [{a_{11}}(x)u + {a_{12}}(x)v],\quad {L_2}v = \mu [{a_{12}}(x)u + {a_{22}}(x)v]\] in a smooth bounded domain $\Omega \subseteq {{\mathbf {R}}^N}$, where ${L_1}$ and ${L_2}$ are formally self-adjoint 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 $({\ast })$, i.e., the set $(\lambda , \mu ) \in {{\mathbf {R}}^2}: ({\ast })$ 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.