Dirk Pauly
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View article: Shape Derivatives of the Eigenvalues of the De Rham Complex for Lipschitz Deformations and Variable Coefficients: Part II
Shape Derivatives of the Eigenvalues of the De Rham Complex for Lipschitz Deformations and Variable Coefficients: Part II Open
In this second part of our series of papers, we develop an abstract framework suitable for de Rham complexes that depend on a parameter belonging to an arbitrary Banach space. Our primary focus is on spectral perturbation problems and the …
View article: Shape Derivatives of the Eigenvalues of the De Rham Complex for Lipschitz Deformations and Variable Coefficients: Part I
Shape Derivatives of the Eigenvalues of the De Rham Complex for Lipschitz Deformations and Variable Coefficients: Part I Open
We study eigenvalue problems for the de Rham complex on varying three dimensional domains. Our analysis includes the Helmholtz equation as well as the Maxwell system with mixed boundary conditions and non-constant coefficients. We provide …
View article: Families of annihilating skew-selfadjoint operators and their connection to Hilbert complexes
Families of annihilating skew-selfadjoint operators and their connection to Hilbert complexes Open
In this short note we show that Hilbert complexes are strongly related to what we shall call annihilating sets of skew-selfadjoint operators. This provides for a new perspective on the classical topic of Hilbert complexes viewed as familie…
View article: Hilbert complexes with mixed boundary conditions part 3: Biharmonic complexes
Hilbert complexes with mixed boundary conditions part 3: Biharmonic complexes Open
We show that the biharmonic Hilbert complex with mixed boundary conditions on bounded strong Lipschitz domains is closed and compact. The crucial results are compact embeddings that follow by abstract arguments using functional analysis to…
View article: Weak equals strong L2 regularity for partial tangential traces on Lipschitz domains
Weak equals strong L2 regularity for partial tangential traces on Lipschitz domains Open
We investigate the boundary trace operators that naturally correspond to $\mathrm{H}(\operatorname{curl},Ω)$, namely the tangential and twisted tangential trace, where $Ω\subseteq \mathbb{R}^{3}$. In particular we regard partial tangential…
View article: Families of Annihilating Skew-Selfadjoint Operators and their Connection to Hilbert Complexes
Families of Annihilating Skew-Selfadjoint Operators and their Connection to Hilbert Complexes Open
In this short note we show that Hilbert complexes are strongly related to what we shall call annihilating sets of skew-selfadjoint operators. This provides for a new perspective on the classical topic of Hilbert complexes viewed as familie…
View article: Traces for Hilbert complexes
Traces for Hilbert complexes Open
We study a new notion of trace operators and trace spaces for abstract Hilbert complexes. We introduce trace spaces as quotient spaces/annihilators. We characterize the kernels and images of the related trace operators and discuss duality …
View article: Hilbert complexes with mixed boundary conditions—Part 2: Elasticity complex
Hilbert complexes with mixed boundary conditions—Part 2: Elasticity complex Open
We show that the elasticity Hilbert complex with mixed boundary conditions on bounded strong Lipschitz domains is closed and compact. The crucial results are compact embeddings which follow by abstract arguments using functional analysis t…
View article: Hilbert Complexes with Mixed Boundary Conditions -- Part 3: Biharmonic Complexes
Hilbert Complexes with Mixed Boundary Conditions -- Part 3: Biharmonic Complexes Open
We show that the biharmonic Hilbert complex with mixed boundary conditions on bounded strong Lipschitz domains is closed and compact. The crucial results are compact embeddings which follow by abstract arguments using functional analysis t…
View article: Traces for Hilbert Complexes
Traces for Hilbert Complexes Open
We study a new notion of trace operators and trace spaces for abstract Hilbert complexes. We introduce trace spaces as quotient spaces/annihilators. We characterize the kernels and images of the related trace operators and discuss duality …
View article: The index of some mixed order Dirac type operators and generalised Dirichlet–Neumann tensor fields
The index of some mixed order Dirac type operators and generalised Dirichlet–Neumann tensor fields Open
We revisit a construction principle of Fredholm operators using Hilbert complexes of densely defined, closed linear operators and apply this to particular choices of differential operators. The resulting index is then computed using an exp…
View article: Hilbert complexes with mixed boundary conditions part 1: de Rham complex
Hilbert complexes with mixed boundary conditions part 1: de Rham complex Open
We show that the de Rham Hilbert complex with mixed boundary conditions on bounded strong Lipschitz domains is closed and compact. The crucial results are compact embeddings which follow by abstract arguments using functional analysis toge…
View article: Hilbert Complexes with Mixed Boundary Conditions -- Part 2: Elasticity Complex
Hilbert Complexes with Mixed Boundary Conditions -- Part 2: Elasticity Complex Open
We show that the elasticity Hilbert complex with mixed boundary conditions on bounded strong Lipschitz domains is closed and compact. The crucial results are compact embeddings which follow by abstract arguments using functional analysis t…
View article: Hilbert Complexes with Mixed Boundary Conditions: Regular Decompositions, Compact Embeddings, and Functional Analysis ToolBox -- Part 1: De Rham Complex
Hilbert Complexes with Mixed Boundary Conditions: Regular Decompositions, Compact Embeddings, and Functional Analysis ToolBox -- Part 1: De Rham Complex Open
We show that the de Rham Hilbert complex with mixed boundary conditions on
bounded strong Lipschitz domains is closed and compact. The crucial results are
compact embeddings which follow by abstract arguments using functional analysis
toge…
View article: A Compactness Result for the div-curl System with Inhomogeneous Mixed Boundary Conditions for Bounded Lipschitz Domains and Some Applications
A Compactness Result for the div-curl System with Inhomogeneous Mixed Boundary Conditions for Bounded Lipschitz Domains and Some Applications Open
For a bounded Lipschitz domain with Lipschitz interface we show the following compactness theorem: Any $L^2$-bounded sequence of vector fields with $L^2$-bounded rotations and $L^2$-bounded divergences as well as $L^2$-bounded tangential t…
View article: The Index of Some Mixed Order Dirac-Type Operators and Generalised Dirichlet-Neumann Tensor Fields
The Index of Some Mixed Order Dirac-Type Operators and Generalised Dirichlet-Neumann Tensor Fields Open
We revisit a construction principle of Fredholm operators using Hilbert complexes of densely defined, closed linear operators and apply this to particular choices of differential operators. The resulting index is then computed with the hel…
View article: The Elasticity Complex.
The Elasticity Complex. Open
We investigate the Hilbert complex of elasticity involving spaces of symmetric tensor fields. For the involved tensor fields and operators we show closed ranges, Friedrichs/Poincare type estimates, Helmholtz type decompositions, regular de…
View article: The Elasticity Complex: Compact Embeddings and Regular Decompositions
The Elasticity Complex: Compact Embeddings and Regular Decompositions Open
We investigate the Hilbert complex of elasticity involving spaces of symmetric tensor fields. For the involved tensor fields and operators we show closed ranges, Friedrichs/Poincare type estimates, Helmholtz type decompositions, regular de…
View article: Low Frequency Asymptotics and Electro-Magneto-Statics for Time-Harmonic Maxwell's Equations in Exterior Weak Lipschitz Domains with Mixed Boundary Conditions
Low Frequency Asymptotics and Electro-Magneto-Statics for Time-Harmonic Maxwell's Equations in Exterior Weak Lipschitz Domains with Mixed Boundary Conditions Open
We prove that the time-harmonic solutions to Maxwell's equations in a 3D exterior domain converge to a certain static solution as the frequency tends to zero. We work in weighted Sobolev spaces and construct new compactly supported replace…
View article: Friedrichs/Poincare Type Constants for Gradient, Rotation, and Divergence: Theory and Numerical Experiments.
Friedrichs/Poincare Type Constants for Gradient, Rotation, and Divergence: Theory and Numerical Experiments. Open
We give some theoretical as well as computational results on Laplace and Maxwell constants. We consider mixed boundary conditions and bounded Lipschitz domains of arbitrary topology.
View article: 3. Weck’s selection theorem: The Maxwell compactness property for bounded weak Lipschitz domains with mixed boundary conditions in arbitrary dimensions
3. Weck’s selection theorem: The Maxwell compactness property for bounded weak Lipschitz domains with mixed boundary conditions in arbitrary dimensions Open
It is proved that the space of differential forms with weak exterior and co-derivative, is compactly embedded into the space of square integrable differential forms. Mixed boundary conditions on weak Lipschitz domains are considered. Furth…
View article: 10. Time-harmonic electro-magnetic scattering in exterior weak Lipschitz domains with mixed boundary conditions
10. Time-harmonic electro-magnetic scattering in exterior weak Lipschitz domains with mixed boundary conditions Open
This paper treats the time-harmonic electro-magnetic scattering or radiation problem governed by Maxwell's equations in an exterior weak Lipschitz domain divided into two disjoint weak Lipschitz parts We will present a solution theory usin…
View article: Solution Theory, Variational Formulations, and Functional a Posteriori Error Estimates for General First Order Systems with Applications to Electro-Magneto-Statics and More
Solution Theory, Variational Formulations, and Functional a Posteriori Error Estimates for General First Order Systems with Applications to Electro-Magneto-Statics and More Open
We prove a comprehensive solution theory using tools from functional analysis, show corresponding variational formulations, and present functional a posteriori error estimates for general linear first order systems. As a prototypical appli…
View article: A global div-curl-lemma for mixed boundary conditions in weak Lipschitz domains and a corresponding generalized A 0 * \mathrm{A}_{0}^{*} - A 1 \mathrm{A}_{1} -lemma in Hilbert spaces
A global div-curl-lemma for mixed boundary conditions in weak Lipschitz domains and a corresponding generalized A 0 * \mathrm{A}_{0}^{*} - A 1 \mathrm{A}_{1} -lemma in Hilbert spaces Open
We prove global and local versions of the so-called {\operatorname{div}} - {\operatorname{curl}} -lemma, a crucial result in the homogenization theory of partial differential equations, for mixed boundary conditions on bounded weak Lipschi…
View article: The Stationary Stokes Problem in Exterior Domains: Estimates of the Distance to Solenoidal Fields and Functional A Posteriori Error Estimates
The Stationary Stokes Problem in Exterior Domains: Estimates of the Distance to Solenoidal Fields and Functional A Posteriori Error Estimates Open
This paper is concerned with the analysis of the inf-sup condition arising in the stationary Stokes problem in exterior domains. We deduce values of the constant in the stability lemma, which yields fully computable estimates of the distan…