Spectral properties of the operator bundles generated by elliptic boundary-value problems in a cone

Abstract

SPECTRAL PROPERTIES OF THE OPERATOR BUNDLES GENERATED BY ELLIPTIC BOUNDARY-VALUE PROBLEMS IN A CONE V. A. Kozlov and V. G. Maz'ya UDC 517 As is known [i], the asymptotics of solutions of elliptic boundary-value problems near conic points of the boundary is determined by the eigenvalues of the boundary-value problems polynomially depending on a spectral parameter, in .regions of the unit sphere. Knowing the particulars of the location of these eigenvalues, one can draw conclusions about singulari- ties, continuity, or smoothness of the solutions; therefore this information is of interest for applications. No general, even to some extent, methods of obtaining this information are known, since even for the simplest problems of mathematical physics these spectral prob- lems have a rather complicated form. The first boundary-value problem for the systems of Lam~ and Stokes and for the biharmonic equation was studied from the indicated viewpoint in [2, 3], where estimates of the width of a strip in the complex plane, free of the spectrum of the corresponding operator bundles, were obtained. Let us also point out [4], where the asymptotics was found of small eigenvalues of the operator bundle generated by the Dirichlet problem for the general elliptic equation of arbitrary order in the exterior of a slender cone. Nonetheless, the questions under discussion are far from settled. For example, boundary- value problems other than the first one have not been studied. As an example of an unsettled question, we can point out the problem of mathematical validation of the fact, obvious to specialists in theoretical mechanics, that a solution of a Lam~ system is continuous at the vertex of a cone, if the boundary is free of stresses near the vertex. In the present paper, we investigate spectral properties of the operator bundles, in a region of a sphere, associated with the boundary-value problems for strongly elliptic sys- tems of order 2m in an n-dimensional cone. It is assumed that the surface of the cone is explicitly expressible in the Cartesian coordinate system. In the case of the first boundary-value problem, we show that the strip {%~C: llm ~--