arXiv subjects

Janusz Ginster

Publications and source records attributed to Janusz Ginster.

17 recordsLinked to original sources

arXiv2025

On the formation of microstructure and the occurrence of vortices in a singularly perturbed energy related to helimagnetism: a scaling law result

In this work, singularly perturbed energies arising from discrete $J_1$-$J_3$-models are studied. The energies under consideration consist of a non-convex bulk term and a higher-order regularizing term and are subject to incompatible boundary conditions. In contrast to existing results in the literature, in this work, admissible fields are not necessarily gradient fields, instead their curl is linked to topological singularities, so-called vortices, in the discrete $J_1$-$J_3$-model. The main result of this work is a scaling law for the minimal energy with respect to three parameters: one measuring the incompatibility of the boundary conditions, the second measuring the strength of the regularizing term, and the third being related to the interatomic distance in the discrete model. The shown result implies in particular that in certain parameter regimes, minimizers necessarily develop vortices. A key tool in the analysis is a careful modification of the celebrated ball-construction technique that, due to a lack of rigidity, considers simultaneously both the bulk energy and the regularizing term.

math.AP↗

arXiv2025

Derivation of the Reissner-Mindlin model from nonlinear elasticity

We discuss how the Reissner-Mindlin plate model can be derived from three-dimensional finite elasticity in terms of $Γ$-convergence. The presence of transverse shear effects in the Reissner-Mindlin model requires to scale different components of the three-dimensional elastic strain differently. A main technical tool is then the combination of rigidity estimates for the deformation and suitably averaged versions.

math.AP↗

arXiv2025

A convex variational principle for the necessary conditions of classical optimal control

A scheme for generating a family of convex variational principles is developed, the Euler- Lagrange equations of each member of the family formally corresponding to the necessary conditions of optimal control of a given system of ordinary differential equations (ODE) in a well-defined sense. The scheme is applied to the Quadratic-Quadratic Regulator problem for which an explicit form of the functional is derived, and existence of minimizers of the variational principle is rigorously shown. It is shown that the Linear-Quadratic Regulator problem with time-dependent forcing can be solved within the formalism without requiring any nonlinear considerations, in contrast to the use of a Riccati system in the classical methodology. Our work demonstrates a pathway for solving nonlinear control problems via convex optimization.

math.OC↗

arXiv2024

Variational Dual Solutions of Chern-Simons Theory

A scheme for generating weakly lower semi-continuous action functionals corresponding to the Euler-Lagrange equations of Chern-Simons theory is described. Coercivity is deduced for such a functional in appropriate function spaces to prove the existence of a minimizer, which constitutes a solution to the Euler-Lagrange equations of Chern-Simons theory in a relaxed sense. A geometric analysis is also made, especially for the gauge group SU(2), relating connection forms on the bundle to corresponding forms in the dual scheme.

math-ph↗

arXiv2024

Nonlinear interpolation inequalities with fractional Sobolev norms and pattern formation in biomembranes

We consider a one-dimensional version of a variational model for pattern formation in biological membranes. The driving term in the energy is a coupling between the order parameter and the local curvature of the membrane. We derive scaling laws for the minimal energy. As a main tool we present new nonlinear interpolation inequalities that bound fractional Sobolev seminorms in terms of a Cahn-Hillard/Modica-Mortola energy.

math.AP↗

arXiv2024

A Hidden Convexity of Nonlinear Elasticity

A technique for developing convex dual variational principles for the governing PDE of nonlinear elastostatics and elastodynamics is presented. This allows the definition of notions of a variational dual solution and a dual solution corresponding to the PDEs of nonlinear elasticity, even when the latter arise as formal Euler-Lagrange equations corresponding to non-quasiconvex elastic energy functionals whose energy minimizers do not exist. This is demonstrated rigorously in the case of elastostatics for the Saint-Venant Kirchhoff material (in all dimensions), where the existence of variational dual solutions is also proven. The existence of a variational dual solution for the incompressible neo-Hookean material in 2-d is also shown. Stressed and unstressed elastostatic and elastodynamic solutions in 1 space dimension corresponding to a non-convex, double-well energy are computed using the dual methodology. In particular, we show the stability of a dual elastodynamic equilibrium solution for which there are regions of non-vanishing length with negative elastic stiffness, i.e.~non-hyperbolic regions, for which the corresponding primal problem is ill-posed and demonstrates an explosive `Hadamard instability;' this appears to have implications for the modeling of physically observed softening behavior in macroscopic mechanical response.

math.AP↗

arXiv2024

Microstructures in a two-dimensional frustrated spin system: Scaling regimes and a discrete-to-continuum limit

We study pattern formation within the $J_1$-$J_3$ - spin model on a two-dimensional square lattice in the case of incompatible (ferromagnetic) boundary conditions on the spin field. We derive the discrete-to-continuum $Γ$-limit at the helimagnetic/ferromagnetic transition point, which turns out to be characterized by a singularly perturbed multiwell energy functional on gradient fields. Furthermore, we study the scaling law of the discrete minimal energy. The constructions used in the upper bound include besides rather uniform or complex branching-type patterns also structures with vortices. Our results show in particular that in certain parameter regimes the formation of vortices is energetically favorable.

math.AP↗

arXiv2024

On the Effect of Geometry on Scaling Laws for a Class of Martensitic Phase Transformations

We study scaling laws for singular perturbation problems associated with a class of two-dimensional martensitic phase transformations and deduce a domain dependence of the scaling law in the singular perturbation parameter. In these settings the respective scaling laws give rise to a selection principle for specific, highly symmetric domain geometries for the associated nucleation microstructure. More precisely, firstly, we prove a general lower bound estimate illustrating that in settings in which the domain and well geometry are incompatible in the sense of the Hadamard-jump condition, then necessarily at least logarithmic losses in the singular perturbation parameter occur in the associated scaling laws. Secondly, for specific phase transformations in two-dimensional settings we prove that this gives rise to a dichotomy involving logarithmic losses in the scaling law for generic domains and optimal linear scaling laws for very specific, highly compatible polygonal domains. In these situations the scaling law thus gives important insight into optimal isoperimetric domains. We discuss both the geometrically linearized and nonlinear settings.

math.AP↗

arXiv2024

A scaling law for a model of epitaxially strained elastic films with dislocations

A static variational model for shape formation in heteroepitaxial crystal growth is considered. The energy functional takes into account surface energy, elastic misfit-energy and nucleation energy of dislocations. A scaling law for the infimal energy is proven. The results quantify the expectation that in certain parameter regimes, island formation or topological defects are favorable. This generalizes results in the purely elastic setting from [Goldman and Zwicknagl (2014)]. To handle dislocations in the lower bound, a new variant of a ball-construction combined with thorough local estimates is presented.

math.AP↗

arXiv2023

The Euler-Bernoulli limit of thin brittle linearized elastic beams

We show that the linear brittle Griffith energy on a thin rectangle $Γ$-converges after rescaling to the linear one-dimensional brittle Euler-Bernoulli beam energy. In contrast to the existing literature, we prove a corresponding sharp compactness result, namely a suitable weak convergence after subtraction of piecewise rigid motions with the number of jumps bounded by the energy

math.AP↗

arXiv2022

An anisotropic Poincaré inequality in $GSBV^p$ and the limit of strongly anisotropic Mumford-Shah functionals

We show that functions in $GSBV^p$ in three-dimensional space with small variation in $2$ of $3$ directions are close to a function of one variable outside an exceptional set. Bounds on the volume and the perimeter in these two directions of the exceptional sets are provided. As a key tool we prove an approximation result for such functions by functions in $W^{1,p}$. For this we present a two-dimensional countable ball construction that allows to carefully remove the jumps of the function. As a direct application, we show $Γ$-convergence of an anisotropic three-dimensional Mumford-Shah model to a one-dimensional model.

math.AP↗

arXiv2020

Rotations with Constant Curl are Constant

It is a classical result that if $u \in C^2(\mathbb{R}^n;\mathbb{R}^n)$ and $\nabla u \in SO(n)$ it follows that $u$ is rigid. In this article this result is generalized to matrix fields with non-vanishing curl. It is shown that every matrix field $R\in C^2(Ω\subseteq \mathbb{R}^3;SO(3))$ such that $\operatorname{curl } R = constant$ is necessarily constant. Moreover, it is proved in arbitrary dimensions that a measurable rotation field is as regular as its distributional curl allows. In particular, a measurable matrix field $R: Ω\to SO(n)$, whose curl in the sense of distributions is smooth, is also smooth.

math.AP↗

arXiv2020

On the motion of curved dislocations in three dimensions: Simplified linearized elasticity

It is shown that in core-radius cutoff regularized simplified elasticity (where the elastic energy depends quadratically on the full displacement gradient rather than its symmetrized version), the force on a dislocation curve by the negative gradient of the elastic energy asymptotically approaches the mean curvature of the curve as the cutoff radius converges to zero. Rigorous error bounds in Hölder spaces are provided. As an application, convergence of dislocations moving by the gradient flow of the elastic energy to dislocations moving by the gradient flow of the arclength functional, when the motion law is given by an $H^1$-type dissipation, and convergence to curve shortening flow in co-dimension $2$ for the usual $L^2$-dissipation is established. In the second scenario, existence and regularity are assumed while the $H^1$-gradient flow is treated in full generality (for short time). The methods developed here are a blueprint for the more physical setting of linearized isotropic elasticity.

math.AP↗

arXiv2018

Solvation in the Large Box Limit

In this paper, the authors study the limit of a sharp interface model for the solvation of charged molecules in an implicit solvent as the number of solute molecules and the size of the surrounding box tend to infinity. The energy is given by a combination of local terms accounting for the physical presence of the molecules in the solvent and a nonlocal electrical energy with or without an ionic effect. In the presence of an ionic effect, the authors prove a screening effect in the limit, i.e., the limit is completely localized and hence electrical long-range interactions of the molecules can be neglected. In the absence of the ionic effect, the authors show that the behavior of the energy depends on the scaling of the number of molecules with respect to the size of the surrounding box. All scaling regimes are identified and corresponding limit results proved. In regimes with many solute molecules this limit includes electrical interactions of $H^{-1}$-type between the molecules.

math.AP↗

arXiv2018

Strain-Gradient Plasticity as the $Γ$-Limit of a Nonlinear Dislocation Energy with Mixed Growth

In this paper a we derive by means of $Γ$-convergence a macroscopic strain-gradient plasticity from a semi-discrete model for dislocations in an infinite cylindrical crystal. In contrast to existing work, we consider an energy with subquadratic growth close to the dislocations. This allows to treat the stored elastic energy without the need to introduce an ad-hoc cut-off radius. As the main tool to prove a complementing compactness statement, we present a generalized version of the geometric rigidity result for fields with non-vanishing $\operatorname{curl}$. A main ingredient is a fine decomposition result for $L^1$-functions whose divergence is in certain critical Sobolev spaces.

math.AP↗

arXiv2018

Plasticity as the $Γ$-Limit of a Two-Dimensional Dislocation Energy: the Critical Regime without the Assumption of Well-Separateness

In this paper, a strain-gradient plasticity model is derived from a mesoscopic model for straight parallel edge dislocations in an infinite cylindrical crystal. The main difference to existing work is that in this work the well-separateness of disloactions is not assumed. In order to prove meaningful lower bounds the ball construction technique, which was developed in the context of Ginzburg-Landau by Jerrard and Sandier, is adapted and modified. To overcome the difficulty of a loss of rigidity on thin annuli during the ball construction a combination of combinatorial arguments and local modifications of the occurring elastic strains is presented.

math.AP↗

arXiv2015

A $BV$ Functional and its Relaxation for Joint Motion Estimation and Image Sequence Recovery

The estimation of motion in an image sequence is a fundamental task in image processing. Frequently, the image sequence is corrupted by noise and one simultaneously asks for the underlying motion field and a restored sequence. In smoothly shaded regions of the restored image sequence the brightness constancy assumption along motion paths leads to a pointwise differential condition on the motion field. At object boundaries which are edge discontinuities both for the image intensity and for the motion field this condition is no longer well defined. In this paper a total-variation type functional is discussed for joint image restoration and motion estimation. This functional turns out not to be lower semicontinuous, and in particular fine-scale oscillations may appear around edges. By the general theory of vector valued $BV$ functionals its relaxation leads to the appearance of a singular part of the energy density, which can be determined by the solution of a local minimization problem at edges. Based on bounds for the singular part of the energy and under appropriate assumptions on the local intensity variation one can exclude the existence of microstructures and obtain a model well-suited for simultaneous image restoration and motion estimation. Indeed, the relaxed model incorporates a generalized variational formulation of the brightness constancy assumption. The analytical findings are related to ambiguity problems in motion estimation such as the proper distinction between foreground and background motion at object edges.

math.AP↗