arXiv subjects

Stefan Krömer

Publications and source records attributed to Stefan Krömer.

At least 19 recordsLinked to original sources

arXiv2026

On the single field formulation in magnetostatics

We systematically discuss the equivalence of two variational formulations of magnetostatics, in terms of magnetization and magnetic field on the one hand and the single field formulation using only magnetic induction. To demonstrate that this link is stable also when the magnetic laws are coupled with other variational static models, elasticity is included in the models as well. Interestingly, despite the fact that the corresponding magnetoelastic energy densities in the material can be computed via Legendre-Fenchel transform in the magnetic state variables, the two formulations are not linked by standard convex duality on the level of the functionals. In addition, convexity and coercivity of the given functional are neither required for the transformation nor always preserved by it.

math-ph↗

arXiv2025

Homogenization of magnetoelastic materials with rigid magnetic inclusions at small strains

We investigate a homogenization problem for a linearly elastic magnetic material that incorporates elastically rigid magnetic inclusions firmly bonded to the matrix. By considering a periodic arrangement of this material, we identify an effective magnetoelastic energy, obtained by homogenization when the period approaches zero. For comparison, we also briefly discuss alternative, essentially equivalent magnetic models naturally linked by a Legendre-Fenchel transform of magnetic energy density where the elastic deformation enters as a parameter.

math.AP↗

arXiv2023

A new example for the Lavrentiev phenomenon in Nonlinear Elasticity

We present a new example for the Lavrentiev phenomenon in context of nonlinear elasticity, caused by an interplay of the elastic energy's resistance to infinite compression and the Ciarlet-Nečas condition, a constraint preventing global interpenetration of matter on sets of full measure.

math.AP↗

arXiv2023

Asymptotic analysis of single-slip crystal plasticity in the limit of vanishing thickness and rigid elasticity

We perform via $Γ$-convergence a 2d-1d dimension reduction analysis of a single-slip elastoplastic body in large deformations. Rigid plastic and elastoplastic regimes are considered. In particular, we show that limit deformations can essentially freely bend even if subjected to the most restrictive constraints corresponding to the elastically rigid single-slip regime. The primary challenge arises in the upper bound where the differential constraints render any bending without incurring an additional energy cost particularly difficult. We overcome this obstacle with suitable non-smooth constructions and prove that a Lavrentiev phenomenon occurs if we artificially restrict our model to smooth deformations. This issue is absent if the differential constraints are appropriately softened.

math.AP↗

arXiv2023

Surface penalization of self-interpenetration in linear and nonlinear elasticity

We analyze a term penalizing surface self-penetration, as a soft constraint for models of hyperelastic materials to approximate the Ciarlet-Nečas condition (almost everywhere global invertibility of deformations). For a linear elastic energy subject to an additional local invertibility constraint, we prove that the penalized elastic functionals converge to the original functional subject to the Ciarlet-Nečas condition. The approach also works for nonlinear models of non-simple materials including a suitable higher order term in the elastic energy, without artificial local constraints. Numerical experiments illustrate our results for a self-contact problem in 3d.

math.AP↗

arXiv2022

Nonlinear elasticity with vanishing nonlocal self-repulsion

We prove that that for nonlinear elastic energies with strong enough energetic control of the outer distortion of admissible deformations, almost everywhere global invertibility as constraint can be obtained in the $Γ$-limit of the elastic energy with an added nonlocal self-repulsion term with asymptocially vanishing coefficient. The self-repulsion term considered here formally coincides with a Sobolev-Slobodeckiĭ seminorm of the inverse deformation. Variants near the boundary or on the surface of the domain are also studied.

math.AP↗

arXiv2021

Relaxation of functionals with linear growth: interactions of emerging measures and free discontinuities

For an integral functional defined on functions $(u,v)\in W^{1,1}\times L^1$ featuring a prototypical strong interaction term between $u$ and $v$, we calculate its relaxation in the space of functions with bounded variations and Radon measures. Interplay between measures and discontinuities bring various additional difficulties, and concentration effects in recovery sequences play a major role for the relaxed functional even if the limit measures are absolutely continuous with respect to the Lebesgue one.

math.AP↗

arXiv2020

Global invertibility for orientation-preserving Sobolev maps via invertibility on or near the boundary

By a result of John Ball (1981), a locally orientation preserving Sobolev map is almost everywhere globally invertible whenever its boundary values admit a homeomorphic extension. As shown here for any dimension, the conclusions of Ball's theorem and related results can be reached while completely avoiding the problem of homeomorphic extension. For suitable domains, it is enough to know that the trace is invertible on the boundary or can be uniformly approximated by such maps. An application in Nonlinear Elasticity is the existence of homeomorphic minimizers with finite distortion whose boundary values are not fixed. As a tool in the proofs, strictly orientation-preserving maps and their global invertibility properties are studied from a purely topological point of view.

math.AP↗

arXiv2019

Quasistatic viscoelasticity with self-contact at large strains

The frame-indifferent viscoelasticity in Kelvin-Voigt rheology at large strains is formulated in the reference configuration (i.e. using the Lagrangian approach) considering also the possible self-contact in the actual deformed configuration. Using the concept of 2nd-grade nonsimple materials, existence of certain weak solutions which are a.e. injective is shown by converging an approximate solution obtained by the implicit time discretisation.

math.AP↗

arXiv2019

Global injectivity in second-gradient Nonlinear Elasticity and its approximation with penalty terms

We present a new penalty term approximating the Ciarlet-Nečas condition (global invertibility of deformations) as a soft constraint for hyperelastic materials. For non-simple materials including a suitable higher order term in the elastic energy, we prove that the penalized functionals converge to the original functional subject to the Ciarlet-Nečas condition. Moreover, the penalization can be chosen in such a way that all low energy deformations, self-interpenetration is completely avoided even for sufficiently small finite values of the penalization parameter. We also present numerical experiments in 2d illustrating our theoretical results.

math.AP↗

arXiv2017

Weak lower semicontinuity by means of anisotropic parametrized measures

It is well known that besides oscillations, sequences bounded only in $L^1$ can also develop concentrations, and if the latter occurs, we can at most hope for weak$^*$ convergence in the sense of measures. Here we derive a new tool to handle mutual interferences of an oscillating and concentrating sequence with another weakly converging sequence. We introduce a couple of explicit examples showing a variety of possible kinds of behavior and outline some applications in Sobolev spaces.

math.AP↗

arXiv2016

Generalized $\mathbf{W^{1,1}}$-Young measures and relaxation of problems with linear growth

We completely characterize generalized Young measures generated by sequences of gradients of maps from $W^{1,1}(Ω;\R^M)$ where $Ω\subset\R^N$. This extends and completes previous analysis by Kristensen and Rindler where concentrations of the sequence of gradients at the boundary of $Ω$ were excluded. We apply our results to relaxation of non-quasiconvex variational problems with linear growth at infinity. We also link our characterization to Souček spaces \cite{soucek}, an extension of $W^{1,1}(Ω;\R^M)$ where gradients are considered as measures on $\barΩ$.

math.AP↗

arXiv2015

Heterogeneous thin films: Combining homogenization and dimension reduction with directors

We analyze the asymptotic behavior of a multiscale problem given by a sequence of integral functionals subject to differential constraints conveyed by a constant-rank operator with two characteristic length scales, namely the film thickness and the period of oscillating microstructures, by means of $Γ$-convergence. On a technical level, this requires a subtile merging of homogenization tools, such as multiscale convergence methods, with dimension reduction techniques for functionals subject to differential constraints. One observes that the results depend critically on the relative magnitude between the two scales. Interestingly, this even regards the fundamental question of locality of the limit model, and, in particular, leads to new findings also in the gradient case.

math.AP↗

arXiv2015

$\mathcal{A}$-quasiconvexity and weak lower semicontinuity of integral functionals

We state necessary and sufficient conditions for weak lower semicontinuity of $u\mapsto\int_Ωh(x,u(x))\,d x$ where $|h(x,s)|\le C(1+|s|^p)$ is continuous and possesses a recession function, and $u\in L^p(Ω;\mathbb{R}^m)$, $p>1$, lives in the kernel of a constant-rank first-order differential operator $\mathcal{A}$ which admits an extension property. Our newly defined notion coincides for $\mathcal{A}=\operatorname{curl}$ with quasiconvexity at the boundary due to J.M. Ball and J. Marsden. Moreover, we give an equivalent condition for weak lower semicontinuity of the above functional along sequences weakly converging in $L^p(Ω;\mathbb{R}^m)$ and approaching the kernel of $\mathcal{A}$ even if $\mathcal{A}$ does not have the extension property.

math.AP↗

arXiv2014

Boundary effects and weak$^*$ lower semicontinuity for signed integral functionals on $\mathrm{BV}$

We characterize lower semicontinuity of integral functionals with respect to weak$^*$ convergence in $\mathrm{BV}$, including integrands whose negative part has linear growth. In addition, we allow for sequences without a fixed trace at the boundary. In this case, both the integrand and the shape of the boundary play a key role. This is made precise in our newly found condition -- quasi-sublinear growth from below at points of the boundary -- which compensates for possible concentration effects generated by the sequence. Our work extends some recent results by J. Kristensen and F. Rindler (Arch. Rat. Mech. Anal. 197 (2010), 539--598 and Calc. Var. 37 (2010), 29--62).

math.AP↗

arXiv2011

Oscillations and concentrations up to the boundary

Oscillations and concentrations in sequences of gradients $\{\nabla u_k\}$, bounded in $L^p(Ω;\R^{M\times N})$ if $p>1$ and $Ω\subset\R^n$ is a bounded domain with the extension property in $W^{1,p}$, and their interaction with local integral functionals can be described by a generalization of Young measures due to DiPerna and Majda. We characterize such DiPerna-Majda measures, thereby extending a result by Kałamajska and Kruž\'ık (2008), where the full characterization was possible only for sequences subject to a fixed Dirichlet boundary condition. As an application we state a relaxation result for noncoercive multiple-integral functionals.

math.AP↗

arXiv2010

A weak-strong convergence property and symmetry of minimizers of constrained variational problems in $\mathbb{R}^N$

We prove a weak-strong convergence result for functionals of the form $\int_{\mathbb{R}^N} j(x, u, Du)\,dx$ on $W^{1,p}$, along equiintegrable sequences. We will then use it to study cases of equality in the extended Polya-Szegö inequality and discuss applications of such a result to prove the symmetry of minimizers of a class of variational problems including nonlocal terms under multiple constraints.

math.AP↗

arXiv2010

Dimension reduction for functionals on solenoidal vector fields

We study integral functionals constrained to divergence-free vector fields in $L^p$ on a thin domain, under standard $p$-growth and coercivity assumptions, $1<p<\infty$. We prove that as the thickness of the domain goes to zero, the Gamma-limit with respect to weak convergence in $L^p$ is always given by the associated functional with convexified energy density wherever it is finite. Remarkably, this happens despite the fact that relaxation of nonconvex functionals subject to the limiting constraint can give rise to a nonlocal functional as illustrated in an example.

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