Fredholm and index theory for symmetrizable hyperbolic systems with nonlocal boundary conditions

Abstract

n this thesis we investigate the Fredholm conditions and the index theory of symmetrizable hyperbolic systems ∂tu = Lu with nonlocal boundary conditions stated on a finite time subset M[0,1] = Σ × [0, 1] of a globally hyperbolic manifold M = Σ × R with boundary components Σ0 = Σ × {0} and Σ1 = Σ × {1}. For two pseudodifferential projections P+,0, P−,1 and two zero order pseudodifferential matrices A0,A1 we consider ∂tu = Lu together with the conditions P+,0A0u(0) = g0 and P−,1A1u(1) = g1 for functions g0 ∈ Im(P+,0) and g1 ∈ Im(P−,1). We derive the general Fredholm conditions for the problem and show that for the cases where the conjugation of P−,0 =: 1 − P+,0 by the solution operator Φ1 to ∂tu = Lu is equal to P−,1 (up to a compact error), the Fredholm conditions can be reduced to the ellipticity of a matrix of G-operators, as long as some assumptions about the group G and the operator L are made. We also apply the results from the abstract Fredholm theory we achieved to the case of the wave equation ∂2 t u = −Δu with a time dependent Laplacian Δ subject to the boundary conditions A0u(0) + B0(∂tu)(0) = g0 ∈ L2(Σ0), and A1u(1) + B1(∂tu)(1) = g1 ∈ L2(Σ1) (with zero order operators B0/1 and first order operators A0/1). The Fredholm conditions for this application of the abstract theory are expressed explicitly as conditions on the operators A0/1 and B0/1 and some special cases are considered, where the index formulas of the problem are given by the Fedosov index formula or some simple trace formula.