arXiv subjects

Heiner Olbermann

Publications and source records attributed to Heiner Olbermann.

At least 19 recordsLinked to original sources

arXiv2025

Asymptotic meshes from $r$-variational adaptation methods for static problems in one dimension

We consider the minimization of integral functionals in one dimension and their approximation by $r$-adaptive finite elements. Including the grid of the FEM approximation as a variable in the minimization, we are able to show that the optimal grid configurations have a well-defined limit when the number of nodes in the grid is being sent to infinity. This is done by showing that the suitably renormalized energy functionals possess a limit in the sense of $\Gamma$-convergence. We provide numerical examples showing the closeness of the optimal asymptotic mesh obtained as a minimizer of the $\Gamma$-limit to the optimal finite meshes.

math.NA↗

arXiv2024

Interacting phase fields yielding phase separation on surfaces

In the present article we study diffuse interface models for two-phase biomembranes. We will do so by starting off with a diffuse interface model on $\mathbb{R}^n$ defined by two coupled phase fields $u,v$. The first phase field $u$ is the diffuse approximation of the interior of the membrane; the second phase field $v$ is the diffuse approximation of the two phases of the membrane. We prove a compactness result and a lower bound in the sense of $\Gamma$-convergence for pairs of phase functions $(u_\varepsilon,v_\varepsilon)$. As an application of this first result, we consider a diffuse approximation of a two-phase Willmore functional plus line tension energy.

math.AP↗

arXiv2024

Connecting disclinations by ridges

We consider a thin elastic sheet with a finite number of disclinations in a variational framework in the F\"oppl-von K\'arm\'an approximation. Under the non-physical assumption that the out-of-plane displacement is a convex function, we prove that minimizers display ridges between the disclinations. We prove the associated energy scaling law with upper and lower bounds that match up to logarithmic factors in the thickness of the sheet. One of the key estimates in the proof that we consider of independent interest is a generalization of the monotonicity property of the Monge-Amp\`ere measure.

math.AP↗

arXiv2023

Phase separation on varying surfaces and convergence of diffuse interface approximations

In this paper we consider phase separations on (generalized) hypersurfaces in Euclidian space. We consider a diffuse surface area (line tension) energy of Modica-Mortola type and prove a compactness and lower bound estimate in the sharp interface limit. We use the concept of generalized BV functions over currents as introduced by Anzellotti et. al. [Annali di Matematica Pura ed Applicata, 170, 1996] to give a suitable formulation in the limit and achieve the necessary compactness property. We also consider an application to phase separated biomembranes where a Willmore energy for the membranes is combined with a generalized line tension energy. For a diffuse description of such energies we give a lower bound estimate in the sharp interface limit.

math.AP↗

arXiv2023

Consistent and convergent discretizations of Helfrich-type energies on general meshes

We show that integral curvature energies on surfaces of the type $E_0(M) := \int_M f(x,n_M(x),D n_M(x))\,d\mathcal{H}^2(x)$ have discrete versions for triangular complexes, where the shape operator $D n_M$ is replaced by the piecewise gradient of a piecewise affine edge director field. We combine an ansatz-free asymptotic lower bound for any uniform approximation of a surface with triangular complexes and a recovery sequence consisting of any regular triangulation of the limit sequence and an almost optimal choice of edge director.

math.AP↗

arXiv2022

Godunov variables and convex entropy for relativistic fluid dynamics with bulk viscosity

Based on the conservation-dissipation formalism proposed by Zhu and collaborators we formulate a general version of the Israel-Stewart theory for relativistic fluid dynamics with bulk viscosity. Our generalization consists in allowing for a wide range of dependence of the entropy density on the bulk viscosity. We show the existence of Godunov-Boillat variables for this model. By known properties of systems possessing such variables, this provides an alternative proof of the recently established existence of solutions for the Israel-Stewart theory locally in time, and a proof that entropy production is positive across weak Lax shocks.

math.AP↗

arXiv2021

Approximation of the Willmore energy by a discrete geometry model

We prove that a certain discrete energy for triangulated surfaces, defined in the spirit of discrete differential geometry, converges to the Willmore energy in the sense of $Γ$-convergence. Variants of this discrete energy have been discussed before in the computer graphics literature.

math.AP↗

arXiv2020

Michell truss type theories as a $Γ$-limit of optimal design in linear elasticity

We show how to derive (variants of) Michell truss theory in two and three dimensions rigorously as the vanishing weight limit of optimal design problems in linear elasticity in the sense of $Γ$-convergence. We improve our previous results in that our treatment here includes the three dimensional case and that we allow for more general boundary conditions and applied forces.

math.AP↗

arXiv2019

Coarea formulae and chain rules for the Jacobian determinant in fractional Sobolev spaces

We prove weak and strong versions of the coarea formula and the chain rule for distributional Jacobian determinants $Ju$ for functions $u$ in fractional Sobolev spaces $W^{s,p}(Ω)$, where $Ω$ is a bounded domain in $\mathbb{R}^n$ with smooth boundary. The weak forms of the formulae are proved for the range $sp>n-1$, $s> \frac{n-1}{n}$, while the strong versions are proved for the range $sp\geq n$, $s\geq \frac{n}{n+1}$. We also provide a chain rule for distributional Jacobian determinants of Hölder functions and point out its relation to two open problems in geometric analysis.

math.AP↗

arXiv2019

Variational competition between full Hessian and its determinant for convex functions

We prove upper and lower bounds for a variational functional for convex functions satisfying certain boundary conditions on a sector of the unit ball in two dimensions. The functional contains two terms: The full Hessian and its determinant, where the former is treated as a small perturbation in the space $L^2$ and the latter as the leading-order term, in the negative Sobolev space $W^{-2,2}$. We point out how this setting is motivated by problems in nonlinear elasticity, and obtain a corollary for a variational problem based on the so-called F\"oppl-von-K\'arm\'an energy.

math.AP↗

arXiv2019

On a $Γ$-limit of Willmore functionals with additional curvature penalization term

We consider the Willmore functional on graphs, with an additional penalization of the area where the curvature is non-zero. Interpreting the penalization parameter as a Lagrange multiplier, this corresponds to the Willmore functional with a constraint on the area where the graph is flat. Sending the penalization parameter to $\infty$ and rescaling suitably, we derive the limit functional in the sense of $Γ$-convergence.

math.AP↗

arXiv2018

On a boundary value problem for conically deformed thin elastic sheets

We consider a thin elastic sheet in the shape of a disk that is clamped at its boundary such that the displacement and the deformation gradient coincide with a conical deformation with no stretching there. We define the free elastic energy as a variation of the von Kármán energy, that penalizes bending energy in $L^p$ with $p\in (2,\frac83)$ (instead of, as usual, $p=2$). We prove ansatz free upper and lower bounds for the elastic energy that scale like $h^{p/(p-1)}$, where $h$ is the thickness of the sheet.

math.AP↗

arXiv2017

Michell trusses in two dimensions as a Gamma-limit of optimal design problems in linear elasticity

We reconsider the minimization of the compliance of a two dimensional elastic body with traction boundary conditions for a given weight. It is well known how to rewrite this optimal design problem as a nonlinear variational problem. We take the limit of vanishing weight by sending a suitable Lagrange multiplier to infinity in the variational formulation. We show that the limit, in the sense of $Γ$-convergence, is a certain Michell truss problem. This proves a conjecture by Kohn and Allaire.

math.AP↗

arXiv2017

The shape of low energy configurations of a thin elastic sheet with a single disclination

We consider a geometrically fully nonlinear variational model for thin elastic sheets that contain a single disclination. The free elastic energy contains the thickness $h$ as a small parameter. We give an improvement of a recently proved energy scaling law, removing the next-to leading order terms in the lower bound. Then we prove the convergence of (almost-)minimizers of the free elastic energy towards the shape of a radially symmetric cone, up to Euclidean motions, weakly in the spaces $W^{2,2}(B_1\setminus B_ρ;\mathbb{R}^3)$ for every $0<ρ<1$, as the thickness $h$ is sent to 0.

math.AP↗

arXiv2016

Extrinsic curvature of codimension one isometric immersions with Hölder continuous derivatives

We prove that if $n$ is even, $(M,g)$ is a compact $n$-dimensional Riemannian manifold whose Pfaffian form is a positive multiple of the volume form, and $y\in C^{1,α}(M;\mathbb{R}^{n+1})$ is an isometric immersion with $n/(n+1)< α\leq 1$, then $y(M)$ is a surface of bounded extrinsic curvature. This is proved by showing that extrinsic curvature, defined by a suitable pull-back of the volume form on the $n$-sphere via the Gauss map, is identical to intrinsic curvature, defined by the Pfaffian form. This latter fact is stated in form of an integral identity for the Brouwer degree of the Gauss map, that is classical for $C^2$ functions, but new for $n>2$ in the present context of low regularity.

math.DG↗

arXiv2016

Integrability of the Brouwer degree for irregular arguments

We prove that the Brouwer degree $\mathrm{deg}(u,U,\cdot)$ for a function $u\in C^{0,α}( U;\mathbb{R}^n)$ is in $L^p(\mathbb{R}^n)$ if $1\leq p<\frac{nα}d$, where $U\subset \mathbb{R}^n$ is open and bounded and $d$ is the box dimension of $\partial U$. This is supplemented by a theorem showing that $u_j\to u$ in $C^{0,α}(U;\mathbb{R}^n)$ implies $\mathrm{deg}(u_j,U,\cdot)\to \mathrm{deg}(u,U,\cdot)$ in $L^p(\mathbb{R}^n)$ for the parameter regime $1\leq p<\frac{nα}d$, while there exist convergent sequences $u_j\to u$ in $C^{0,α}(U;\mathbb{R}^n)$ such that $\|\mathrm{deg}(u_j,U,\cdot)\|_{L^p}\to \infty$ for the opposite regime $p>\frac{nα}d$.

math.CA↗

arXiv2015

Symmetry breaking in indented elastic cones

Motivated by simulations of carbon nanocones (see Jordan and Crespi, Phys. Rev. Lett., 2004), we consider a variational plate model for an elastic cone under compression in the direction of the cone symmetry axis. Assuming radial symmetry, and modeling the compression by suitable Dirichlet boundary conditions at the center and the boundary of the sheet, we identify the energy scaling law in the von-Kármán plate model. Specifically, we find that three different regimes arise with increasing indentation $δ$: initially the energetic cost of the logarithmic singularity dominates, then there is a linear response corresponding to a moderate deformation close to the boundary of the cone, and for larger $δ$ a localized inversion takes place in the central region. Then we show that for large enough indentations minimizers of the elastic energy cannot be radially symmetric. We do so by an explicit construction that achieves lower elastic energy than the minimum amount possible for radially symmetric deformations.

math.AP↗

arXiv2015

Energy scaling law for a single disclination in a thin elastic sheet

We consider a single disclination in a thin elastic sheet of thickness $h$. We prove ansatz-free lower bounds for the free elastic energy in three different settings: First, for a geometrically fully non-linear plate model, second, for three-dimensional nonlinear elasticity, and third, for the Föppl-von Kármán plate theory. The lower bounds in the first and third result are optimal in the sense that we find upper bounds that are identical to the respective lower bounds in the leading order of $h$.

math.AP↗