We are interested in nonlocal boundary value problems associated with group actions on manifolds. In these problems, both the main operator and the boundary operator belong to operator algebras generated by pseudodifferential operators on the manifold and so-called 'shift operators', Fourier integral operators associated with diffeomorphisms of the base space.
Nonlocal problems of this type arise in concrete applications (plasma physics, theory of multi-layer plates and shells used in aviation and astronautics, in optical systems with two-dimensional feedback, etc.), but they are also interesting from the point of view of noncommutative geometry, because their symbols form noncommutative algebras. We intend to study elliptic and hyperbolic nonlocal boundary value problems associated with such group actions. Our aim is
(i) to investigate the analytic aspects of the theory, in
particular ellipticity and the Fredholm property
(ii) to establish
index formulae and
(iii) to use semiclassical methods to obtain asymptotics for
hyperbolic problems.
This is part of a joint German-Russian project with Anton Savin (RUDN, Moscow) and Vladimir Nazaikinkii (Russian Academy of Sciences).
Publications
We consider a smooth compact manifold with boundary, M, embedded in a smooth manifold of the same dimension on which an amenable group \(\Gamma\) acts by isometries. We do not assume M to be invariant under \(\Gamma\). This results in a partial action of \(\Gamma\) on: For $g\in \Gamma$ we let \(M^\circ_g = g(M^\circ)\cap M^\circ\) and obtain diffeomorphisms \(g:M^\circ_{g^{-1}} \to
M^\circ_g
\).
We assume that any two images of \(\partial M\) under \(\Gamma\) either coincide or are disjoint and that only finitely many lie in M. The spherical blow-up of these images of \(\partial M\) in M yields a
manifold Y with boundary consisting of finitely many components. Moreover, Y inherits a partial action of \(\Gamma\).
We can then define the C*-algebra \(\mathcal
A=\overline{\Psi_\Gamma(Y,\partial Y)}\) of operators on \(L^2(Y)\oplus
L^2(\partial Y)\), generated by the algebra \(\Psi(Y,\partial Y)\) of operators of order and type zero in Boutet de Monvel's calculus on Y and partial isometries associated with the partial action. Denote by
\(\Sigma=\overline{\Psi(Y,\partial Y)}/\mathcal K
\) the symbol space. If the partial action of \(\Gamma\) on Prim(\(\Sigma\)) is topologically
free, we find a criterion for the Fredholm property of the operators in \(\overline{\Psi_\Gamma(Y,\partial Y)}\).
Moreover, we obtain the classification of the elliptic elements in
\(\overline{\Psi_\Gamma(Y,\partial Y)}\) modulo stable homotopies: For \(\mathcal A_0= C(Y\sqcup \partial Y)\rtimes\Gamma\)
\(Ell(\mathcal A_0,\mathcal A)\cong K_0(C_0(T^*Y^\circ)\rtimes\Gamma
)\oplus K_0(C(\partial Y)\rtimes \Gamma).\)
If \(\Gamma\) is finitely generated and of polynomial growth, then the elements associated with the second summand do not contribute to the index.
Related project(s):
80Nonlocal boundary problems: Index theory and semiclassical asymptotics
Let G be a compact Lie group that acts smoothly on a closed manifold M. Using a general Simonenko principle, we derive a novel criterion for the Fredholm property of G-pseudodifferential operators acting on Sobolev spaces of sections of vector bundles over M. In case the group is finite, we obtain a further characterization of the Fredholm property of G-pseudodifferential operators in terms of the invertibility of suitable symbols.
Related project(s):
80Nonlocal boundary problems: Index theory and semiclassical asymptotics
We study the Fredholm solvability for a new class of nonlocal boundary value problems associated with group actions on smooth manifolds. Namely, we consider the case in which the group action is defined on an ambient manifold without boundary and does not preserve the manifold with boundary on which the problem is stated. In particular, the group action does not map the boundary to itself. The orbits of the boundary under the group action split the manifold into subdomains, and this decomposition, being combined with the C*-algebra techniques, plays an important role in our approach to the analysis of the problem.
| Journal | Math. Notes |
| Volume | 111 no. 5-6 |
| Pages | 701-721 |
| Link to preprint version | |
| Link to published version |
Related project(s):
80Nonlocal boundary problems: Index theory and semiclassical asymptotics
Team Members
Dr. Eske Ewert
Researcher
Leibniz-Universität Hannover
Prof. Dr. Elmar Schrohe
Project leader
Leibniz-Universität Hannover