arXiv subjects

Nathanael Skrepek

Publications and source records attributed to Nathanael Skrepek.

14 recordsLinked to original sources

arXiv2026

Nonlinear Media via Nonlocal Homogenisation

We consider a nonlinear PDE describing a nonlinear electrostatic medium with nonlocal dielectricity. The existence proof for the corresponding equation is based on Schauder's theorem and a new compactness theorem for moving coefficients (``Helga's Theorem''). This technique uses insights from (operator-theoretic/topological) homogenisation theory. Surprisingly, even though monotonicity assumptions are neither used nor valid, the underlying domain is only required to be weak Lipschitz and no assumption on the derivatives of the nonlinearity is needed.

math.AP↗

arXiv2025

Well-posedness and stability of the Lagrange representation of the n-D wave equation via boundary triples

We study the Lagrange representation of the wave equation with generalized Laplacian $\operatorname{div} T \nabla$. We allow the coefficients -- the Young modulus $T$ and the density $ρ$ -- to be $\mathrm{L}^{\infty}$ or even nonlocal operators. Moreover, the Lipschitz boundary of the domain $Ω$ can be split into several parts admitting Dirichlet, Neumann and/or Robin-boundary conditions of displacement, velocity and stress. We show well-posedness of this classical model of the wave equation utilizing boundary triple theory for skew-adjoint operators. In addition we show semi-uniform stability of solutions under slightly stronger assumptions by means of a spectral result.

math.AP↗

arXiv2025

Analysis of coupled Maxwell-cable problems

Building on the recently published work "Modeling of radiating curved cables via coupled telegrapher's and Maxwell's equations", which introduces a model for the interaction between electromagnetic fields and radiating (possibly curved) cables, we analyze the qualitative properties of the resulting dynamical system. The model features inputs and outputs given by the currents and voltages at the cable ends, while the state comprises the corresponding distributions along the cables and the electromagnetic fields in the surrounding domain. We show that the autonomous dynamics (i.e., with zero input) generate a strongly continuous semigroup and establish sufficient conditions for well-posedness, meaning continuous dependence of the state and output trajectories on the inputs and initial conditions.

math.AP↗

arXiv2025

Modeling of radiating curved cables via coupled telegrapher's and Maxwell's equations

We investigate the electromagnetic interactions of cable harnesses in the time domain. We present a novel model that allows for curved cables, extending the standard assumptions typically made in transmission line modeling. The cables are described by the telegrapher's equations, the classical model for transmission lines, driven by input signals implemented through appropriate boundary conditions, such as imposed voltages at cable ends. The cables interact via electromagnetic radiation; the latter is determined by Maxwell's equations. This interaction is incorporated into the model through boundary conditions imposed on the electromagnetic field. The resulting coupling between the transmission lines and Maxwell's equations is energetically consistent. In particular, we show that the coupled system satisfies a global power balance.

math.AP↗

arXiv2025

Homogenisation for Maxwell and Friends

We refine the understanding of continuous dependence on coefficients of solution operators under the nonlocal $H$-topology viz Schur topology in the setting of evolutionary equations in the sense of Picard. We show that certain components of the solution operators converge strongly. The weak convergence behaviour known from homogenisation problems for ordinary differential equations is recovered on the other solution operator components. The results are underpinned by a rich class of examples that, in turn, are also treated numerically, suggesting a certain sharpness of the theoretical findings. Analytic treatment of an example that proves this sharpness is provided too. Even though all the considered examples contain local coefficients, the main theorems and structural insights are of operator-theoretic nature and, thus, also applicable to nonlocal coefficients. The main advantage of the problem class considered is that they contain mixtures of type, potentially highly oscillating between different types of PDEs; a prototype can be found in Maxwell's equations highly oscillating between the classical equations and corresponding eddy current approximations.

math.AP↗

arXiv2025

Weak equals strong L2 regularity for partial tangential traces on Lipschitz domains

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 traces, i.e., we look only on a subset $Γ$ of the boundary $\partialΩ$. We assume both $Ω$ and $Γ$ to be strongly Lipschitz (possibly unbounded). We define the space of all $\mathrm{H}(\operatorname{curl},Ω)$ fields that possess a $\mathrm{L}^{2}$ tangential trace in a weak sense and show that the set of all smooth fields is dense in that space, which is a generalization of Belgacem, Bernardi, Costabel and Dauge 1997. This is especially important for Maxwell's equation with mixed boundary condition as we answer the open problem by Weiss and Staffans 2013 (Section 5) for strongly Lipschitz pairs.

math.FA↗

arXiv2023

Semi-uniform stabilization of anisotropic Maxwell's equations via boundary feedback on split boundary

We regard anisotropic Maxwell's equations as a boundary control and observation system on a bounded Lipschitz domain. The boundary is split into two parts: one part with perfect conductor boundary conditions and the other where the control and observation takes place. We apply a feedback control law that stabilizes the system in a semi-uniform manner without any further geometric assumption on the domain. This will be achieved by separating the equilibriums from the system and show that the remaining system is described by an operator with compact resolvent. Furthermore, we will apply a unique continuation principle on the resolvent equation to show that there are no eigenvalues on the imaginary axis.

math.AP↗

arXiv2023

Characterizations of the Sobolev space $\mathrm{H}^{1}$ on the boundary of a strong Lipschitz domain in 3-D

In this work we investigate the Sobolev space $\mathrm{H}^{1}(\partialΩ)$ on a strong Lipschitz boundary $\partialΩ$, i.e., $Ω$ is a strong Lipschitz domain. In most of the literature this space is defined via charts and Sobolev spaces on flat domains. We show that there is a different approach via differential operators on $Ω$ and a weak formulation directly on the boundary that leads to the same space. This second characterization of $\mathrm{H}^{1}(\partialΩ)$ is in particular of advantage, when it comes to traces of $\mathrm{H}(\operatorname{curl},Ω)$ vector fields.

math.FA↗

arXiv2023

Weak Operator Continuity for Evolutionary Equations

Considering evolutionary equations in the sense of Picard, we identify a certain topology for material laws rendering the solution operator continuous if considered as a mapping from the material laws into the set of bounded linear operators, where the latter are endowed with the weak operator topology. The topology is a topology of vector-valued holomorphic functions and provides a lift of the previously introduced nonlocal $\mathrm{H}$-topology to particular holomorphic functions. The main area of applications are nonlocal homogenisation results for coupled systems of time-dependent partial differential equations. A continuous dependence result for a nonlocal model for cell migration is also provided.

math.AP↗

arXiv2023

Quasi Gelfand triples

We generalize the notion of Gelfand triples (also called Banach-Gelfand triples or rigged Hilbert spaces) by dropping the necessity of a continuous embedding. This means in our setting we lack of a chain inclusion. We replace the continuous embedding by a closed embedding of a dense subspace. This new notion will be called quasi Gelfand triple. These triples appear naturally, when we regard the boundary spaces of spatially multidimensional differential operators, e.g., the Maxwell operator. We will show that there is a smallest space where we can continuously embed the entire triple. Moreover, we will show density results for intersections of members of the quasi Gelfand triple. Finally, we show that every quasi Gelfand triple can be decomposed into two "ordinary" Gelfand triples.

math.FA↗

arXiv2022

Mixed-Dimensional Geometric Coupling of Port-Hamiltonian Systems

We propose a new interconnection relation for infinite-dimensional port-Hamiltonian systems that enables the coupling of ports with different spatial dimensions by integrating over the the surplus dimensions. To show the practical relevance, we apply this interconnection to a model system of an actively cooled gas turbine blade. We also show that this interconnection relation behaves well with respect to a discretization in finite element space, ensuring its usability for practical applications.

math.DS↗

arXiv2021

Stability of the multidimensional wave equation in port-Hamiltonian modelling

We investigate the stability of the wave equation with spatial dependent coefficients on a bounded multidimensional domain. The system is stabilized via a scattering passive feedback law. We formulate the wave equation in a port-Hamiltonian fashion and show that the system is semi-uniform stable, which is a stability concept between exponential stability and strong stability. Hence, this also implies strong stability of the system. In particular, classical solutions are uniformly stable. This will be achieved by showing that the spectrum of the port-Hamiltonian operator is contained in the left half plane $\mathbb{C}_{-}$ and the port-Hamiltonian operator generates a contraction semigroup. Moreover, we show that the spectrum consists of eigenvalues only and the port-Hamiltonian operator has a compact resolvent.

math.FA↗

arXiv2021

A Compactness Result for the div-curl System with Inhomogeneous Mixed Boundary Conditions for Bounded Lipschitz Domains and Some Applications

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 traces on one part of the boundary and $L^2$-bounded normal traces on the other part of the boundary, contains a strongly $L^2$-convergent subsequence. This generalises recent results for homogeneous mixed boundary conditions by the first author and collaborators. As applications we present a related Friedrichs/Poincare type estimate, a div-curl lemma, and show that the Maxwell operator with mixed tangential and impedance boundary conditions (Robin type boundary conditions) has compact resolvents.

math.AP↗

arXiv2019

Well-posedness of linear first order Port-Hamiltonian Systems on multidimensional spatial domains

We consider a port-Hamiltonian system on a spatial domain $Ω\subseteq \mathbb{R}^n$ that is bounded with Lipschitz boundary. We show that there is a boundary triple associated to this system. Hence, we can characterize all boundary conditions that provide unique solutions that are non-increasing in the Hamiltonian. As a by-product we develop the theory of quasi Gelfand triples. Adding ``natural'' boundary controls and boundary observations yields scattering/impedance passive boundary control systems. This framework can be applied to the wave equation, Maxwell equations and Mindlin plate model, and probably many more.

math.FA↗