Boundary-value Problems for the Squared Laplace Operator

The squared Laplace operator acting on symmetric rank-two tensor fields is studied on a (flat) Riemannian manifold with smooth boundary. Symmetry of this fourth-order elliptic operator is obtained provided that such tensor fields and their first (or second) normal derivatives are set to zero at the boundary. Strong ellipticity of the resulting boundary-value problems is also proved. Mixed boundary conditions are eventually studied which involve complementary projectors and tangential differential operators. In such a case, strong ellipticity is guaranteed if a pair of matrices are non-degenerate. These results find application to the analysis of quantum field theories on manifolds with boundary. PACS 03.70-Theory of quantized fields. PACS 04.60-Quantum gravity. 1 1.-Introduction Mathematicians and physicists have become familiar, along the years, with the key role played by the operators of Dirac and Laplace type in the investigation of elliptic operators on Riemannian manifolds. For example, it is by now well known that the symbol of the Dirac operator is a generator of all elliptic symbols on (closed) Riemannian manifolds [1], and deep results in index theory have been found by looking at non-local boundary conditions for operators of Dirac type [1,2]. More recently, local supersymmetry and quantum supergravity have led to the consideration of local boundary conditions for the Dirac operator [1,3–5], while theoretical models relevant for quantum chromodynamics rely on mathematical structures weaker than the standard spin-structures where, again, a Dirac operator is found to be quite essential [1,6]. This analysis leads, in turn, to a deeper understanding of the geometry and topology of four-manifolds [6–8]. The operators of Laplace type, on the other hand, occur naturally in the consideration of quantized gauge theories (Abelian and non-Abelian) and Euclidean quantum gravity [5,9], and many efforts are devoted, within that framework, to the evaluation of the one-loop semiclassical approximation in quantum field theory, with the help of heat-kernel and ζ-function methods [4,5,10,11]. Nevertheless, there is still room left for the analysis of many other classes of differential operators on manifolds. In particular, we are here concerned with the so-called conformally covariant operators [12,13]. They arise in the course of studying the behaviour of field theories and differential geometric objects under conformal rescalings of the background metric g. More precisely, if the conformal rescaling of g is written in the form g ω = e 2ω g, a conformally covariant operator Q satisfies, by definition, the transformation property Q ω = e …