Spectral Theory for Boundary Value Problems for Elliptic Systems of Mixed Order by Giuseppe Geymonat and Gerd Grubb

Introduction. For a closed, densely defined linear operator T in a Hubert space H, we define the essential spectrum ess sp T as the complement in C of the set of X for which T X is a Fredholm operator (with possibly nonzero index). Recall (cf. Wolf [7] ) that X G ess sp T if and only if either T — X or 7* — X has a singular sequence, i.e. a sequence ukGH with \\uk\\ = 1 for all k, (T X)uk —> 0 (or ( 7 * \)uk —> 0) in H, but uk having no convergent subsequence in H. ess sp T is closed and invariant under compact perturbations of T9 and contains the accumulation points of the eigenvalue spectrum. Let £2 be an ^-dimensional compact C°° manifold with boundary V and interior £2 = Û\F. It is well known that when A is a properly elliptic operator on £2 of order r > 0, the L-realization A# : u I—• Au with domain D(A g ) = {u G L (Sl)\ Au G L(Sl), Bu\T = 0}, defined by a boundary operator B that covers A (i.e. {A, B} defines an elliptic boundary value problem), has ess sp A% = 0 . However, when A is a system of mixed order, elliptic in the sense of Douglis and Nirenberg (cf. [1]), ess sp A$ can be nonempty even when {A,B} is elliptic with smooth coefficients and £2 is compact. We study this phenomenon for a class of Douglis-Nirenberg systems of nonnegative order, determine the essential spectrum, and find the asymptotic behavior of the discrete spectrum at + °° for the selfadjoint lower bounded realizations. Examples of the systems we consider are: The linearized Navier-Stokes operator and certain systems stemming from nuclear reactor analysis. A preliminary, less advanced account of the theory was given in [5].