Instytut Matematyczny Polskiej Akademii Nauk / Institute of Mathematics / Publishing house / Banach Center Publications / All volumes

Abstract

Differential operators: the gradient $\operatorname{grad}$ and the divergence $\operatorname{div}$ are defined and examined in the bundles of symmetric tensors on a Riemannian manifold. For the second order operator $\operatorname{div} \operatorname{grad}$ which appears to be elliptic and a manifold with boundary a system of natural boundary conditions is constructed and investigated. There are $2^{k+1}$ conditions in the bundle $S^k$ of symmetric tensors of degree $k$. This is in contrast to the bundle of skew-symmetric forms where (for analogous differential operators) there are always four such conditions independently of the degree of forms (i.e. independently of $k$). All the $2^{k+1}$ conditions are investigated in detail. In particular, it is proved that each of them is self-adjoint and elliptic. Such ellipticity of a given boundary condition has an essential significance for the existing of a discrete spectrum and an orthonormal basis in $L^2$ consisting of smooth sections that are the eigenvectors of the operator and satisfy the boundary condition. Some special cases, e.g. $k=1$ or the cases that the boundary is umbilical or totally geodesic, are also discussed.