On the complex spectra of second order elliptic systems

Abstract

A basic problem in the theory of partial differential equations is that of describing the spectra of elliptic operators. If an operator is self-adjoint or symmetric in the appropriate sense, the spectrum is real; but general elliptic operators may have complex spectrum. The object of this article is to study the complex spectra of second order elliptic systems. If the elliptic system arises from linearization about a steady state of a reaction-diffusion system, information about the signs of the real parts of complex eigenvalues is relevant to questions of stability and bifurcation. We give bounds for the real parts of eigenvalues which prevent them from changing sign, and discuss some examples where complex eigenvalues may cross the imaginary axis as a parameter in the system varies. We also discuss briefly the extent to which the theory of positive operators can be used to give bounds for complex eigenvalues. For the case of a single second order elliptic operator of general form on a bounded domain, the fundamental bounds on the spectrum were obtained by Protter and Weinberger in [17]. They considered the problem n n (1.1 ) Cu == L a;j{x)uZiZ; + L b;{x)uz; + c{x)u = Am{x)u. ';,;=1 ;=1 in 0, u = ° on ao, with (a;j) positive definite and symmetric, c ~ 0, m > 0, and ° ~Rn bounded. They showed that for any smooth function w that is positive on n and any eigenvalue A,