Abstract
AbstractWe 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 using an explicit description 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 [42] 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 Poincaré–Friedrichs type useful for elasticity or the theory of general relativity.
Highlights
Mathematics Subject Classification Primary 47A53 (Fredholm operators; index theories) · 58B15 (Fredholm structures); Secondary 35G15 (Boundary value problems for linear higherorder equations) · 35Q61 (Maxwell equations) · 35Q41 (Time-dependent Schrödinger equations, Dirac equations) · 35Q40 (PDEs in connection with quantum mechanics)
A thorough understanding of the Fredholm case has led to further examples for the Witten index, which in turn might prove useful for both mathematics and physics
Lemmas 5.1, 5.2, and Theorem 5.3 show that the compactness properties, the dimensions of the kernels and cohomology groups, the maximal compactness, and the Fredholm indices of the second biharmonic complex do not depend on the material weights ε and μ
Summary
We employ the notations and assumptions of Sect. 1. ) = ker(curl) ∩ ker(div), ) = ker(CurlS) ∩ ker(divDivS), ) = ker(symCurlT) ∩ ker(DivT), ) = ker(CurlCurlS ) ∩ ker(DivS), HRNhm( HbNih,T,1( HbNih,S,2( HeNla,S(. ) = ker(div) ∩ ker(curl), ) = ker(DivT) ∩ ker(symCurlT), ) = ker(divDivS) ∩ ker(CurlS), ) = ker(DivS) ∩ ker(CurlCurlS ). We will compute the dimensions of the kernels N0, N2,∗, i.e., dim ker(grad) = 0, dim ker(Gradgrad) = 0, dim ker(devGrad) = 0, dim ker(symGrad) = 0, dim ker(grad) = n, dim ker(devGrad) = 4n, dim ker(Gradgrad) = 4n, dim ker(symGrad) = 6n, and the dimensions of the cohomology groups K1, K2, i.e., dim HRDhm( ) = m − 1, dim HRNhm( ) = p, dim HbDih,S,1( dim HbDih,T,2(. Remark 2.1 leads to the following problem that seems to be open: Problem 2.2 Is it possible to find differential operators on ⊆ R3 (bounded, strong Lipschitz domain) of the form (1.1) as discussed in Remark 2.1 that violate the general index formula for D in (2.1)?
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