In dieser Arbeit wird ein neues Existenzresultat für das Stefan-Problem mit Gibbs-Thomson-Problem vorgestellt.Im Unterschied zum Vorgehen von Luckhaus bei seinem Beweis der Existenz und Uneindeutigkeit schwacher Lösungen ...

Ziel ist die Implementierung und Analyse eines effizienten numerischen Verfahrens zur Lösung der Cahn-Hilliard- und der Cahn-Larché-Gleichung. In Kapitel 2 wird das mathematische Modell detailliert beschrieben, und die ...

In this thesis we consider systems of partial differential equations of continuum mechanics and analyze regularity properties of their weak solutions. <br /> The first chapter contains a detailed introduction and reviews the contents of chapter two, three and four. <br /> We start in chapter 2 with the local regularity problem related to the equations modelling the mechanical behaviour of elasto-perfect plastic materials respectively to an elasto-viscoplastic approximation of these materials, i. e. we consider the Norton-Hoff approximation to Hencky's law. These equations form a nonlinear systems of partial differential equations of second order and of elliptic type in the usual primal formulation, where one is interested in the displacement vector u=u(x) respectively the strain tensor ε(u) = ½ (∇u + (∇u)^T). <br /> We study these systems via a dual approach which was developed by A. Bensoussan and J. Frehse. In this approach we look for the stress tensor σ = σ(x) which solves the system of equations: <br />A σ + | σ^D |^p-2 σ^D = ε(u)<br /> div σ + f = 0<br /> in the weak sense with mixed boundary conditions. We show local Hölder continuity of the stress tensor in two dimensions for the Norton-Hoff approximation of the Hencky law in plasticity theory and deduce also corresponding results for the strain tensor ε(u). <br /> The main tool to achieve this result is a logarithmic Morrey estimate, which was developed by J. Frehse together with A. Bensoussan and G. Seregin in the here considered context of the dual theory of elliptic systems. These logarithmic Morrey estimates combined with a suitable adapted estimate on higher integrability a la Meyers-Necas-Gehring-Giaquinta-Modica give the final result. <br /> We also deal with a system of partial differential equations describing a steady motion of an incompressible fluid with shear-dependent viscosity and present a new global existence result for p > 2d / d+2. Here p is the coercivity parameter of the nonlinear elliptic operator related to the stress tensor and d is the dimension of the space. Lipschitz test functions, a subtle splitting of the level sets of the maximal functions for the velocity gradients, and a decomposition of the pressure are incorporated to obtain almost everywhere convergence of the velocity gradients. <br /> Finally we survey and improve some results concering uniqueness and regularity of solutions to the instationary Navier-Stokes equations in three (and higher) dimensions. In particular we shall show that the class of weak solutions which additionally belong to the space L² (0,T;BMO) guarantees uniqueness as well as regularity of the weak solution under consideration. We also discuss the related issue of controlling the blow-up phenomenon of smooth solutions to the Navier-Stokes equations. The method of proof which we present is elementary and depends deeply on the special structure of the nonlinear convective term u · ∇ u of the Navier-Stokes equations together with the property div u = 0; namely the convective term is a div-curl expression and according to Coifman, Lions, Meyer and Semmes it belongs to the Hardy space H¹. This shows that this property respective method of proof is also applicable to other equations in hydrodynamics as for example the Boussinesq equations, the equations of Magneto-Hydrodynamics and the equations of higher grade type fluids....

The aim of these notes is to generalize Laumon's construction [18] of automorphic sheaves corresponding to local systems on a smooth, projective curve C to the case of local systems with indecomposable unipotent ramification ...

In this thesis we study the approximation complexity of the Steiner Tree Problem and related problems as well as foundations in structural complexity theory. The Steiner Tree Problem is one of the most fundamental problems ...

Diese Arbeit beschäftigt sich mit einem neuen Ansatz für das Klassifikationsproblem beim Maschinellen Lernen durch Funktionsrekonstruktion. Es basiert auf dem Zugang des Regularisierungsnetzwerks, aber im Gegensatz zu ...

In der Arbeit werden Lösungsverfahren für partielle Differential- gleichungen vorgestellt, die auf Multiskalen-Ansatzfunktionen (Wavelets) basieren. Zur adaptiven Approximation der numerischen Lösung werden anisotrope ...

In dieser Arbeit werden erstmalig über einen zur Nichtstandardform gehörenden Erzeugendensystem-Ansatz robuste Wavelet-basierte Multiskalen-Löser für allgemeine zweidimensionale stationäre Konvektions-Diffusions-Probleme ...

Die vorliegende Arbeit beschaeftigt sich mit der Entwicklung von Verfahren zur interaktiven graphischen Darstellung (Visualisierung) grosser Datenmengen. Insbesondere betrifft dies adaptiv hierarchische Verfahren zur ...

For schemes which are smooth over a regular base scheme we establish the existence of cycle class maps with values in the corresponding algebraic De Rham cohomology. These maps have all the properties one expects, i.e. ...