Regularity aspects of a higher-order variational approach to the denoising and inpainting of images with TV-type energies

This thesis deals with a certain class of variational problems of higher order that stem from applications in mathematical image processing. The main intention is to study the regularity behavior of minimizers of integral functionals on Sobolev spaces with differentiable energy densities of linear growth approximating the TV-case. Building upon results that were given by Bildhauer, Fuchs, Tietz and Weickert in the first-order case, we treat existence of weakly differentiable, relaxed, dual as well as of classically differentiable solutions under suitable conditions on the model. Our considerations are supplemented with a detailed study of the lower-dimensional cases as well as with a coupling model which offers an alternative approach to the higher-order case.Die vorliegende Arbeit beschäftigt sich mit einer bestimmten Klasse von Variationsproblemen höherer Ordnung, die von Anwendungen in der mathematischen Bildverarbeitung herrühren. Das Hauptaugenmerk liegt dabei auf der Untersuchung des Regularitätsverhaltens der Minimierer von Integral-Funktionalen auf Sobolev-Räumen mit differenzierbaren Energiedichten von linearem Wachstum, die den TV-Fall approximieren. Aufbauend auf Resultaten, die von Bildhauer, Fuchs, Tietz und Weickert im Falle erster Ordnung erbracht wurden, behandeln wir die Existenz von schwach differenzierbaren, relaxierten, dualen sowie von klassisch differenzierbaren Lösungen unter jeweils hinreichenden Voraussetzungen an das Modell. Unsere Betrachtungen werden ergänzt durch eine eingehendere Analyse der niederdimensionalen Fälle sowie eines Kopplungsmodells, das einen alternativen Zugang zum Fall höherer Ordnung bietet

Mathematical foundations of elasticity

[Preface] This book treats parts of the mathematical foundations of three-dimensional elasticity using modern differential geometry and functional analysis. It is intended for mathematicians, engineers, and physicists who wish to see this classical subject in a modern setting and to see some examples of what newer mathematical tools have to contribute

Singular integral operators in Morrey spaces and interior regularity of solutions to systems of linear PDE's

We obtain boundedness in Morrey spaces of singular integral operators with Calderon-Zygmund type kernel of mixed homogeneity. These estimates are used for the study of the interior regularity of the solutions of linear elliptic/parabolic systems. The proved Poincare-type inequality permits to describe the Holder, Morrey, and BMO regularity of the lower-order derivatives of the solutions