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In 1914 Bohr proved that there is an r ∈ (0, 1) such that if a power series converges in the unit disk and its sum has modulus less than 1 then, for |z| < r, the sum of absolute values of its terms is again less than 1. Recently analogous results were obtained for functions of several variables. The aim of this paper is to comprehend the theorem of Bohr in the context of solutions to second order elliptic equations meeting the maximum principle.
We construct an algebra of pseudo-differential boundary value problems that contains the classical Shapiro-Lopatinskij elliptic problems as well as all differential elliptic problems of Dirac type with APS boundary conditions, together with their parametrices. Global pseudo-differential projections on the boundary are used to define ellipticity and to show the Fredholm property in suitable scales of spaces.
Die vorliegende Arbeit beschäftigt sich mit der Charakterisierung von Seismizität anhand von Erdbebenkatalogen. Es werden neue Verfahren der Datenanalyse entwickelt, die Aufschluss darüber geben sollen, ob der seismischen Dynamik ein stochastischer oder ein deterministischer Prozess zugrunde liegt und was daraus für die Vorhersagbarkeit starker Erdbeben folgt. Es wird gezeigt, dass seismisch aktive Regionen häufig durch nichtlinearen Determinismus gekennzeichent sind. Dies schließt zumindest die Möglichkeit einer Kurzzeitvorhersage ein. Das Auftreten seismischer Ruhe wird häufig als Vorläuferphaenomen für starke Erdbeben gedeutet. Es wird eine neue Methode präsentiert, die eine systematische raumzeitliche Kartierung seismischer Ruhephasen ermöglicht. Die statistische Signifikanz wird mit Hilfe des Konzeptes der Ersatzdaten bestimmt. Als Resultat erhält man deutliche Korrelationen zwischen seismischen Ruheperioden und starken Erdbeben. Gleichwohl ist die Signifikanz dafür nicht hoch genug, um eine Vorhersage im Sinne einer Aussage über den Ort, die Zeit und die Stärke eines zu erwartenden Hauptbebens zu ermöglichen.
We prove a theorem on analytic representation of integrable CR functions on hypersurfaces with singular points. Moreover, the behaviour of representing analytic functions near singular points is investigated. We are aimed at explaining the new effect caused by the presence of a singularity rather than at treating the problem in full generality.
A numerical bifurcation analysis of the electrically driven plane sheet pinch is presented. The electrical conductivity varies across the sheet such as to allow instability of the quiescent basic state at some critical Hartmann number. The most unstable perturbation is the two-dimensional tearing mode. Restricting the whole problem to two spatial dimensions, this mode is followed up to a time-asymptotic steady state, which proves to be sensitive to three-dimensional perturbations even close to the point where the primary instability sets in. A comprehensive three-dimensional stability analysis of the two-dimensional steady tearing-mode state is performed by varying parameters of the sheet pinch. The instability with respect to three-dimensional perturbations is suppressed by a sufficiently strong magnetic field in the invariant direction of the equilibrium. For a special choice of the system parameters, the unstably perturbed state is followed up in its nonlinear evolution and is found to approach a three-dimensional steady state.
We consider edge-degenerate families of pseudodifferential boundary value problems on a semi-infinite cylinder and study the behavior of their push-forwards as the cylinder is blown up to a cone near infinity. We show that the transformed symbols belong to a particularly convenient symbol class. This result has applications in the Fredholm theory of boundary value problems on manifolds with edges.
An elliptic theory is constructed for operators acting in subspaces defined via even pseudodifferential projections. Index formulas are obtained for operators on compact manifolds without boundary and for general boundary value problems. A connection with Gilkey's theory of η-invariants is established.
An elliptic theory is constructed for operators acting in subspaces defined via even pseudodifferential projections. Index formulas are obtained for operators on compact manifolds without boundary and for general boundary value problems. A connection with Gilkey's theory of η-invariants is established.