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Boundary Value Problems, Weyl Functions, and Differential Operators

For a symmetric operator or relation A with infinite deficiency indices in a Hilbert space we develop an abstract framework for the description of symmetric and selfadjoint extensions A of A as restrictions of an operator or relation T which is a core of the adjoint A . This concept is applied to second order elliptic partial dierential operators on smooth bounded domains, and a class of elliptic problems with eigenvalue dependent boundary conditions is investigated.

Boundary value problems for elliptic partial dierential operators on bounded domains

The Calderón projection: New definition and applications

A Morse–Smale index theorem for indefinite elliptic systems and bifurcation

We consider a curve of Fredholm pairs of Lagrangian subspaces in a xed Banach space with continuously varying weak symplectic structures. Assuming vanishing index, we obtain intrin- sically a continuously varying splitting of the total Banach space into pairs of symplectic subspaces. Using such decompositions we dene the curve's Maslov index by symplectic reduction to the clas- sical nite-dimensional case. We prove the transitivity of repeated symplectic reductions and obtain the invariance of the Maslov in- dex under symplectic reduction, while recovering all the standard properties of the Maslov index. As an application, we consider curves of elliptic operators which have varying principal symbol, varying maximal domain and are not necessarily of Dirac type. For this class of operator curves, we derive a desuspension spectral ow formula for varying well-posed boundary conditions on manifolds with boundary and obtain the splitting of the spectral ow on partitioned manifolds.

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The Maslov Index in Symplectic Banach Spaces

In this thesis we study non-local boundary value problems for elliptic differential operators on manifolds with smooth boundary. First of all a theory for general elliptic operators is developed. Among other things we prove that all weak solutions to a given boundary value problem are strong ones if and only if the boundary condition satisfies a certain regularity estimate. The boundary conditions considered in this thesis are subject to the following condition. The operator and its commutator with a generator of the Sobolev scale on the boundary are bounded on the full Sobolev scale. In comparison with the existing literature this class of boundary conditions is rather general and covers spectral as well as some non-pseudolocal boundary conditions. As is shown in the appendix, the operators satisfying the above-mentioned commutator condition form a C*-Algebra. This turns out to be crucial for the whole theory. The central object in our regularity theory are the Calderon projection and its range, the Cauchy data space. We give a simplified proof for the fact that the Calderon projection of an arbitrary elliptic differential operator is a classical pseudodifferential operator on the boundary. The most important analytic properties of an extension, regularity, well-posedness and self-adjointness, may be characterised in terms of the pair of spaces formed by the Cauchy data space and the boundary values satisfying the given condition. In the case of pseudodifferential conditions this leads to well-known criteria. For formally self-adjoint operators we show that the set of regular self-adjoint extensions is parametrised by the Fredholm Langrange Grassmannian associated to the Cauchy data space. Then we apply the general results to the most important classes of geometric differential operators, namely Dirac and Laplace operators. We show, e.g., how cobordism invariance of the index of Dirac operators can be deduced from properties of the Cauchy data space using merely elementary Fredholm theory. For Laplace operators we discuss some classical boundary value problems first. Those boundary conditions leading to self-adjoint are described in detail. Moreover we establish a necessary and sufficient criterion for semi-boundedness.

Christian Frey

Papers

On Non-local Boundary Value Problems for Elliptic Operators