arXiv subjects

Hans Knüpfer

Publications and source records attributed to Hans Knüpfer.

At least 19 recordsLinked to original sources

arXiv2025

Well-Posedness and Regularity of the Heat Equation with Robin Boundary Conditions in the Two-Dimensional Wedge

Well-posedness and higher regularity of the heat equation with Robin boundary conditions in an unbounded two-dimensional wedge is established in an $L^{2}$-setting of monomially weighted spaces. A mathematical framework is developed which allows to obtain arbitrarily high regularity without a smallness assumption on the opening angle of the wedge. The challenging aspect is that the resolvent problem exhibits two breakings of the scaling invariance, one in the equation and one in the boundary condition.

math.AP↗

arXiv2025

Gamma-Convergence of Higher-Order Phase Transition Models

We investigate the asymptotic behavior as $\varepsilon \to 0$ of singularly perturbed phase transition models of order $n \geq 2$, given by \begin{align} G_\varepsilon^{λ,n}[u] := \int_I \frac 1\varepsilon W(u) -λ\varepsilon^{2n-3} (u^{(n-1)})^2 + \varepsilon^{2n-1} (u^{(n)})^2 \ dx, \quad u \in W^{n,2}(I), \end{align} where $λ>0$ is fixed, $I \subset \mathbb{R}$ is an open bounded interval, and $W \in C^0(\mathbb{R})$ is a suitable double-well potential. We find that there exists a positive critical parameter depending on $W$ and $n$, such that the $Γ$-limit of $G_\varepsilon^{λ,n}$ with respect to the $L^1$-topology is given by a sharp interface functional in the subcritical regime. The cornerstone for the corresponding compactness property is a novel nonlinear interpolation inequality involving higher-order derivatives, which is based on Gagliardo-Nirenberg type inequalities.

math.AP↗

arXiv2024

Well-posedness of the Stokes equations on a wedge with Navier-slip boundary conditions

We consider the incompressible and stationary Stokes equations on an infinite two-dimensional wedge with non-scaling invariant Navier-slip boundary conditions. We prove well-posedness and higher regularity of the Stokes problem in a certain class of weighted Sobolev spaces. The novelty of this work is the occurrence of two different scalings in the boundary condition, which is not treated so far for the Stokes system in unbounded wedge-type domains. These difficulties are overcome by first constructing a variational solution in a second order weighted Sobolev space and subsequently proving higher regularity up to the tip of the wedge by employing an iterative scheme. We believe that this method can be used for other problems with variational structure and multiple scales.

math.AP↗

arXiv2022

Minimal Energy for Geometrically Nonlinear Elastic Inclusions in Two Dimensions

We are concerned with a variant of the isoperimetric problem, which in our setting arises in a geometrically nonlinear two-well problem in elasticity. More precisely, we investigate the optimal scaling of the energy of an elastic inclusion of a fixed volume for which the energy is determined by a surface and an (anisotropic) elastic contribution. Following ideas from \cite{CS} and \cite{KnuepferKohn-2011}, we derive the lower scaling bound by invoking a two-well rigidity argument and a covering result. The upper bound follows from a well-known construction for a lens-shaped elastic inclusion.

math.AP↗

arXiv2022

Asymptotic shape of isolated magnetic domains

We investigate the energy of an isolated magnetized domain $Ω\subset \mathbb{R}^n$ for $n=2,3$. In non-dimensionalized variables, the energy given by $$ \mathcal{E}(Ω) \ = \ \int_{\mathbb{R}^n} |\nabla χ_Ω| \ dx + \int_{\mathbb{R}^n} |\nabla h_Ω|^2 \ dx $$ penalizes the interfacial area of the domain as well as the energy of the corresponding magnetostatic field. Here, the magnetostatic potential $h_Ω$ is determined by $Δh_Ω= \partial_1 χ_Ω$, corresponding to uniform magnetization within the domain. We consider the macroscopic regime $|Ω| \rightarrow \infty$, in which we derive compactness and $Γ$-limit which is formulated in terms of the cross-sectional area of the anisotropically rescaled configuration. We then give the solutions for the limit problems.

math.AP↗

arXiv2022

$Γ$-limit for a sharp interface model related to pattern formation on biomembranes

We derive a macroscopic limit for a sharp interface version of a model proposed in [29] to investigate pattern formation due to competition of chemical and mechanical forces in biomembranes. We identify sub- and supercrital parameter regimes and show with the introduction of the autocorrelation function that the ground state energy leads to the isoperimetric problem in the subcritical regime, which is interpreted to not form fine scale patterns.

math.AP↗

arXiv2022

Onset of pattern formation in thin ferromagnetic films with perpendicular anisotropy

We consider the onset of pattern formation in an ultrathin ferromagnetic film of the form $\tildeΩ_t := \tildeΩ \times [0,t]$ for $\tilde{ Ω} \Subset \mathbb{R}^2$ with preferred perpendicular magnetization direction. The relative micromagnetic energy is given by \begin{align} \mathcal{E}[M] &= \int_{\tildeΩ_t} d^2 |\nabla M|^2+ Q \int_{\tildeΩ_t} (M_1^2+M_2^2) + \int_{\mathbb{R}^3} |\mathcal{H}(M)|^2 - \int_{\mathbb{R}^3} |\mathcal{H}(e_3 χ_{\tilde{ Ω}})|^2, \end{align} describing the energy difference for a given magnetization $M : \mathbb{R}^3 \to \mathbb{R}^3$ with $|M| = χ_{\tilde{ Ω}_t}$ and the uniform magnetization $e_3 χ_{\tilde{ Ω}_t}$. For $t \ll d$, we establish the scaling of the energy and a BV-bound in the critical regime here the base area of the film is of order $|\tilde{ Ω}| \sim (Q-1)^{1/2} d e^{\frac{2πd}t \sqrt{Q-1}}$. We furthermore investigate the onset of non-trivial pattern formation in the critical regime depending on the size of the rescaled film.

math-ph↗

arXiv2021

Second order expansion for the nonlocal perimeter functional

The seminal results of Bourgain, Brezis, Mironescu and Dávila show that the classical perimeter can be approximated by a family of nonlocal perimeter functionals. We consider a corresponding second order expansion for the nonlocal perimeter functional. In a special case, the considered family of energies is also relevant for a variational model for thin ferromagnetic films. We derive the Gamma--limit of these functionals. We also show existence for minimizers with prescribed volume fraction. For small volume fraction, the unique, up to translation, minimizer of the limit energy is given by the ball. The analysis is based on a systematic exploitation of the associated symmetrized autocorrelation function.

math.AP↗

arXiv2021

$Γ$-limit for two-dimensional charged magnetic zigzag domain walls

Charged domain walls are a type of domain walls in thin ferromagnetic films which appear due to global topological constraints. The non-dimensionalized micromagnetic energy for a uniaxial thin ferromagnetic film with in-plane magnetization $m \in \mathbb{S}^1$ is given by \begin{align*} E_ε[m] \ = \ ε\|\nabla m\|_{L^2}^2 + \frac {1}ε \|m \cdot e_2\|_{L^2}^2 + \frac{πλ}{2|\lnε|} \|\nabla \cdot (m-M)\|_{\dot H^{-\frac{1}{2}}}^2, \end{align*} where magnetization in $e_1$-direction is globally preferred and where $M$ is an arbitrary fixed background field to ensure global neutrality of magnetic charges. We consider a material in the form a thin strip and enforce a charged domain wall by suitable boundary conditions on $m$. In the limit $ε\to 0$ and for fixed $λ> 0$, corresponding to the macroscopic limit, we show that the energy $Γ$-converges to a limit energy where jump discontinuities of the magnetization are penalized anisotropically. In particular, in the subcritical regime $λ\leq 1$ one-dimensional charged domain walls are favorable, in the supercritical regime $λ> 1$ the limit model allows for zigzaging two-dimensional domain walls.

math.AP↗

arXiv2020

On the optimality of the rock-salt structure among lattices with charge distributions

The goal of this work is to investigate the optimality of the $d$-dimensional rock-salt structure, i.e., the cubic lattice $V^{1/d}\mathbb{Z}^d$ of volume $V$ with an alternation of charges $\pm 1$ at lattice points, among periodic distribution of charges and lattice structures. We assume that the charges are interacting through two types of radially symmetric interaction potentials, according to their signs. We first restrict our study to the class of orthorhombic lattices. We prove that, for our energy model, the $d$-dimensional rock-salt structure is always a critical point among periodic structures of fixed density. This holds for a large class of potentials. We then investigate the minimization problem among orthorhombic lattices with an alternation of charges for inverse power laws and Gaussian interaction potentials. High density minimality results and low-density non-optimality results are derived for both types of potentials. Numerically, we investigate several particular cases in dimensions $2$, $3$ and $8$. The numerics support the conjecture that the rock-salt structure is the global optimum among all lattices and periodic charges, satisfying some natural constraints. For $d=2$, we observe a phase transition of the type 'triangular-rhombic-square-rectangular' for the minimizer's shape as the density decreases.

math-ph↗

arXiv2020

Note on crystallization for alternating particle chains

We investigate one-dimensional periodic chains of alternate type of particles interacting through mirror symmetric potentials. The optimality of the equidistant configuration at fixed density -- also called crystallization -- is shown in various settings. In particular, we prove the crystallization at any scale for neutral and non-neutral systems with inverse power laws interactions, including the three-dimensional Coulomb potential. We also show the minimality of the equidistant configuration at high density for systems involving inverse power laws and repulsion at the origin. Furthermore, we derive a necessary condition for crystallization at high density based on the positivity of the Fourier transform of the interaction potentials sum.

math.OC↗

arXiv2018

Emergence of non-trivial minimizers for the three-dimensional Ohta-Kawasaki energy

This paper is concerned with the diffuse interface Ohta-Kawasaki energy in three space dimensions, in a periodic setting, in the parameter regime corresponding to the onset of non-trivial minimizers. We identify the scaling in which a sharp transition from asymptotically trivial to non-trivial minimizers takes place as the small parameter characterizing the width of the interfaces between the two phases goes to zero, while the volume fraction of the minority phases vanishes at an appropriate rate. The value of the threshold is shown to be related to the optimal binding energy solution of Gamow's liquid drop model of the atomic nucleus. Beyond the threshold the average volume fraction of the minority phase is demonstrated to grow linearly with the distance to the threshold. In addition to these results, we establish a number of properties of the minimizers of the sharp interface screened Ohta-Kawasaki energy in the considered parameter regime. We also establish rather tight upper and lower bounds on the value of the transition threshold.

math.AP↗

arXiv2018

On Born's conjecture about optimal distribution of charges for an infinite ionic crystal

We study the problem for the optimal charge distribution on the sites of a fixed Bravais lattice. In particular, we prove Born's conjecture about the optimality of the rock-salt alternate distribution of charges on a cubic lattice (and more generally on a d-dimensional orthorhombic lattice). Furthermore, we study this problem on the two-dimensional triangular lattice and we prove the optimality of a two-component honeycomb distribution of charges. The results holds for a class of completely monotone interaction potentials which includes Coulomb type interactions. In a more general setting, we derive a connection between the optimal charge problem and a minimization problem for the translated lattice theta function.

math-ph↗

arXiv2018

Optimal lattice configurations for interacting spatially extended particles

We investigate lattice energies for radially symmetric, spatially extended particles interacting via a radial potential and arranged on the sites of a two-dimensional Bravais lattice. We show the global minimality of the triangular lattice among Bravais lattices of fixed density in two cases: In the first case, the distribution of mass is sufficiently concentrated around the lattice points, and the mass concentration depends on the density we have fixed. In the second case, both interacting potential and density of the distribution of mass are described by completely monotone functions in which case the optimality holds at any fixed density.

math-ph↗

arXiv2017

Self-similar lifting and persistent touch-down points in the thin-film equation

We study self-similar solutions of the thin-film equation, with mobility exponent m in (0,4], that describe the lifting of an isolated touch-down point given by an initial profile of the form |x|. This provides a mechanism for non-uniqueness of the thin-film equation with m in (2,4), since solutions with a persistent touch-down point also exist in this case. In order to prove existence of the self-similar solutions, we need to study a four-dimensional continuous dynamical system. The proof consists of a shooting argument based on the identification of invariant regions and on suitable energy formulas.

math.DS↗

arXiv2017

Magnetic domains in thin ferromagnetic films with strong perpendicular anisotropy

We investigate the scaling of the ground state energy and optimal domain patterns in thin ferromagnetic films with strong uniaxial anisotropy and the easy axis perpendicular to the film plane. Starting from the full three-dimensional micromagnetic model, we identify the critical scaling where the transition from single domain to multidomain ground states such as bubble or maze patterns occurs. Furthermore, we analyze the asymptotic behavior of the energy in two regimes separated by a transition. In the single domain regime, the energy $Γ$-converges towards a much simpler two-dimensional and local model. In the second regime, we derive the scaling of the minimal energy and deduce a scaling law for the typical domain size.

math.AP↗

arXiv2015

A sharp interface evolutionary model for shape memory alloys

We show the existence of an energetic solution to a quasistatic evolutionary model of shape memory alloys. Elastic behavior of each material phase/variant is described by polyconvex energy density. Additionally, to every phase boundary, there is an interface-polyconvex energy assigned, introduced by M. Šilhavý. The model considers internal variables describing the evolving spatial arrangement of the material phases and a deformation mapping with its first-order gradients. It allows for injectivity and orientation-preservation of deformations. Moreover, the resulting material microstructures have finite length scales.

math.AP↗

arXiv2015

Optimal distribution of oppositely charged phases: perfect screening and other properties

We study the minimum energy configuration of a uniform distribution of negative charge subject to Coulomb repulsive self-interaction and attractive interaction with a fixed positively charged domain. After having established existence and uniqueness of a minimizing configuration, we prove charge neutrality and the complete screening of the Coulomb potential exerted by the positive charge, and we discuss the regularity properties of the solution. We also determine, in the variational sense of $Γ$-convergence, the limit model when the charge density of the negative phase is much higher than the positive one.

math.AP↗