The Index of Some Mixed Order Dirac-Type Operators and Generalised Dirichlet-Neumann Tensor Fields

TL;DR

This paper computes the index of mixed order Dirac-type operators linked to elasticity and biharmonic complexes, describing associated cohomology groups and basis functions with applications in physics and numerical analysis.

Contribution

It introduces a novel approach to compute indices of complex differential operators of mixed order, expanding the theory of Dirac-type operators in elasticity and relativity.

Findings

Computed indices for Dirac-type operators in elasticity and biharmonic complexes.

Explicitly described cohomology groups and basis functions using topological invariants.

Provided new vector-analytical estimates for applications in elasticity and general relativity.

Abstract

We revisit a construction principle of Fredholm operators using Hilbert complexes of densely defined, closed linear operators and apply this to particular choices of differential operators. The resulting index is then computed with the help of explicitly describing the dimension of the cohomology groups of generalised (`harmonic') Dirichlet and Neumann tensor fields. The main results of this contribution are the computation of the indices of Dirac-type operators associated to the elasticity complex and the newly found biharmonic complex, relevant for the biharmonic equation, elasticity, and for the theory of general relativity. The differential operators are of mixed order and cannot be seen as leading order type with relatively compact perturbation. As a by-product we present a comprehensive description of the underlying generalised Dirichlet-Neumann vector and tensor fields defining the respective cohomology groups, including an explicit construction of bases in terms of topological invariants, which are of both analytical and numerical interest. Though being defined by certain projection mechanisms, we shall present a way of computing these basis functions by solving certain PDEs given in variational form. For all of this we rephrase core arguments in the work of Rainer Picard [1982] applied to the de Rham complex and use them as a blueprint for the more involved cases presented here. In passing, we also provide new vector-analytical estimates of generalised Poincare-Friedrichs type useful for elasticity or the theory of general relativity.