Denk, Robert

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Robert
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Preface

2021-11-26, Denk, Robert, Giga, Yoshikazu, Kozono, Hideo, Saal, Jürgen, Simonett, Gieri, Titi, Edriss

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The spin-coating process : analysis of the free boundary value problem

2010, Denk, Robert, Geissert, Matthias, Hieber, Matthias, Saal, Jürgen, Sawada, Okihiro

In this paper, an accurate model for the spin-coating process is presented and investigated from the analytical point of view. More precisely, the spin-coatong process is being described as a one-phase free boundary value problem for Newtonian fluids in the rotational setting. The method presented is based on a transformation of the free boundary value problem to a quasilinear evolution equation on a fixed domain. The keypoint for solving the latter equation will be so-called maximal regularity approach. In order to pursue this one needs to determine the precise regularity classes for the associated inhomogenous linearized equations. This is being achieved by applying the Newton polygon method to the boundary sumbol.

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Bounded H-infty-calculus for pseudodifferential Douglis-Nirenberg systems of mild regularity

2007, Denk, Robert, Saal, Jürgen, Seiler, Jörg

Parameter-ellipticity with respect to a closed subsector of the complex plane for pseudodifferential Douglis-Nirenberg systems is introduced and shown to imply the existence of a bounded H_\infty-calculus in suitable scales of Sobolev, Besov, and Hölder spaces. We also admit non pseudodifferential perturbations. Applications concern systems with coefficients of mild Hölder regularity and the generalized thermoelastic plate equations.

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Lp-theory for a fluid–structure interaction model

2020-10, Denk, Robert, Saal, Jürgen

We consider a fluid–structure interaction model for an incompressible fluid where the elastic response of the free boundary is given by a damped Kirchhoff plate model. Utilizing the Newton polygon approach, we first prove maximal regularity in Lp-Sobolev spaces for a linearized version. Based on this, we show existence and uniqueness of the strong solution of the nonlinear system for small data.

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Bounded H-calculus for pseudo-differential Douglis–Nirenberg systems of mild regularity

2009, Denk, Robert, Saal, Jürgen, Seiler, Jörg

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Theorie und Numerik partieller Diferentialgleichungen

2007, Denk, Robert, Saal, Jürgen

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The spin-coating process : analysis of the free boundary value problem

2011, Denk, Robert, Geissert, Matthias, Hieber, Matthias, Saal, Jürgen, Sawada, Okihiro

In this paper, an accurate model of the spin-coating process is presented and investigated from the analytical point of view. More precisely, the spin-coating process is being described as a one-phase free boundary value problem for Newtonian fluids subject to surface tension and rotational effects. It is proved that for T > 0 there exists a unique, strong solution to this problem in (0, T) belonging to a certain regularity class provided the data and the speed of rotation are small enough in suitable norms. The strategy of the proof is based on a transformation of the free boundary value problem to a quasilinear evolution equation on a fixed domain. The keypoint for solving the latter equation is the so-called maximal regularity approach. In order to pursue in this direction one needs to determine the precise regularity classes for the associated inhomogeneous linearized equations. This is being achieved by applying the Newton polygon method to the boundary symbol.

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Inhomogeneous symbols, the Newton polygon, and maximal Lp-regularity

2008, Denk, Robert, Saal, Jürgen, Seiler, Jörg

We prove a maximal regularity result for operators corresponding to rotation invariant symbols (in space) which are inhomogeneous in space and time. Symbols of this type frequently arise in the treatment of half-space models for (free) boundary-value problems. The result is obtained by extending the Newton polygon approach to variables living in complex sectors and combining it with abstract results on the H∞-calculus and R-bounded operator families. As an application, we derive maximal regularity for the linearized Stefan problem with Gibbs-Thomson correction.

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Skript zur Vorlesung Partielle Differentialgleichungen II : Sommersemester 2007

2007, Denk, Robert, Saal, Jürgen

Evolutionsgleichungen und der operatortheoretische Zugang zu ihrer Lösung stehen im Mittelpunkt dieser Vorlesung. Daher werden in diesem einleitenden Abschnitt einige wichtige partielle Differentialgleichungen als abstrakte Evolutionsgleichungen geschrieben, d.h. als Cauchy-Probleme. Dabei handelt es sich um parabolische wie auch hyperbolische Gleichungen. Weiter werden hier einige wichtige Begriffe und Grundlagen aus der Operatortheorie zitiert, welche in den folgenden Abschnitten verwendet werden. Schreibt man eine partielle Differentialgleichung als abstraktes Cauchyproblem, tauchen in natürlicher Weise Integrale und Ableitungen Banachraum-wertiger Funktionen auf. Der zugehörige (Lebesguesche) Integralbegriff ist der des Bochner-Integrals, welches ebenfalls kurz vorgestellt wird.