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In this short note we review results on equilibrium shapes of minimizers to the sessile drop problem. More precisely, we study the Winterbottom problem and prove that the Winterbottom shape is indeed optimal. The arguments presented here are based on relaxation and the (anisotropic) isoperimetric inequality.

We consider a scalar Fermi-Pasta-Ulam-Tsingou (FPUT) system on a square 2D lattice with a cubic nonlinearity. For such systems the NLS equation can be derived to describe the evolution of an oscillating moving wave packet of small amplitude which is slowly modulated in time and space. We show that this NLS approximation makes correct predictions about the dynamics of the original scalar FPUT system for the strain and the displacement variables., Comment: 17 pages

Here we report the performance of a small scale 4 K pulse tube cryocooler operating with a low input power reaching a minimum temperature of 2.2 K, as well as a cooling capacity of over 240 mW at 4.2 K. The compressor is air cooled and can be supplied by single phase power sockets. With an input power of about 1.3 kW the coefficient of performance reaches values of up to 185 mW/kW, which is among the highest currently reported values for small to medium power pulse tubes. The combination of an oil-free Helium compressor and low maintenance pulse tube cryocooler provides a unique miniaturized, energy efficient and mobile cooling tool for applications at 4 K and below., Comment: 6 pages, 3 figures, conference proceedings of 25th joint CEC-ICMC, July 2023 in Honolulu, Hawaii, USA

We establish a general criterion for the existence of convex sets of fixed shape as, e.g., balls of a given radius, of maximal probability on Banach spaces. We also provide counterexamples showing that their existence my fail even in some common situations.

The purpose of this note is to establish two continuum theories for the bending and torsion of inextensible rods as $\Gamma$-limits of 3D atomistic models. In our derivation we study simultaneous limits of vanishing rod thickness $h$ and interatomic distance $\varepsilon$. First, we set up a novel theory for ultrathin rods composed of finitely many atomic fibres ($\varepsilon\sim h$), which incorporates surface energy and new discrete terms in the limiting functional. This can be thought of as a contribution to the mechanical modelling of nanowires. Second, we treat the case where $\varepsilon\ll h$ and recover a nonlinear rod model $-$ the modern version of Kirchhoff's rod theory., Comment: 28 pages, 1 figure

Starting from a particle system with short-range interactions, we derive a continuum model for the bending, torsion, and brittle fracture of inextensible rods moving in three-dimensional space. As the number of particles tends to infinity, it is assumed that the rod's thickness is of the same order as the interatomic distance. Fracture energy in the $\Gamma$-limit is expressed by an implicit cell formula, which covers different modes of fracture, including (complete) cracks, folds and torsional cracks. In special cases, the cell formula can be significantly simplified. Our approach applies e.g. to atomistic systems with Lennard-Jones-type potentials and is motivated by the research of ceramic nanowires., Comment: 38 pages, 2 figures (examples and explicit calculation of crack energy added)

We develop a general stability analysis for objective structures, which constitute a far reaching generalization of crystal lattice systems. We show that these particle systems, although in general neither periodic nor space filling, allow for the identification of stability constants in terms of representations of the underlying symmetry group and interaction potentials. From our representation results a general computational algorithm to test objective structures for their stability is derived. By way of example we show that our method can be applied to verify the stability of carbon nanotubes with chirality.

We establish discrete Korn type inequalities for particle systems within the general class of objective structures that represents a far reaching generalization of crystal lattice structures. For space filling configurations whose symmetry group is a general space group we obtain a full discrete Korn inequality. For systems with non-trivial codimension our results provide an intrinsic rigidity estimate within the extended dimensions of the structure. As their continuum counterparts in elasticity theory, such estimates are at the core of energy estimates and, hence, a stability analysis for a wide class of atomistic particle systems.

We consider a two phase elastic thin film with soft inclusions subject to bending dominated deformations. The soft (void) phase may comprise asymptotically small droplets within the elastic matrix. We perform a dimension reduction analysis and obtain a novel `Blake-Zisserman-Kirchhoff' model that combines Kirchhoff's classical plate theory with Blake-Zisserman type surface energy contributions at cracks, folds and the boundary of voids.

It is well known that a strictly convex minimand admits at most one minimizer. We prove a partial converse: Let $X$ be a locally convex Hausdorff space and $f \colon X \mapsto \left( - \infty , \infty \right]$ a function with compact sublevel sets and exhibiting some mildly superlinear growth. Then each tilted minimization problem \begin{equation} \label{eq. minimization problem} \min_{x \in X} f(x) - \langle x' , x \rangle_X \end{equation} admits at most one minimizer as $x'$ ranges over $\text{dom} \left( \partial f^* \right)$ if and only if the biconjugate $f^{**}$ is essentially strictly convex and agrees with $f$ at all points where $f^{**}$ is subdifferentiable. We prove this via a representation formula for $f^{**}$ that might be of independent interest.

We introduce and study the mechanical system which describes the dynamics and statics of rigid bodies of constant density floating in a calm incompressible fluid. Since much of the standard equilibrium theory, starting with Archimedes, allows bodies with vertices and edges, we assume the bodies to be convex and take care not to assume more regularity than that implied by convexity. One main result is the (Liapunoff) stability of equilibria satisfying a condition equivalent to the standard 'metacentric' criterion., Comment: 22 pages

We study structural properties and the harmonic analysis of discrete subgroups of the Euclidean group. In particular, we 1. obtain an efficient description of their dual space, 2. develop Fourier analysis methods for periodic mappings on them, and 3. prove a Schur-Zassenhaus type splitting result.

We study an approximation scheme for a variational theory of cohesive fracture in a one-dimensional setting. Here, the energy functional is approximated by a family of functionals depending on a small parameter $0 < \varepsilon \ll 1$ and on two fields: the elastic part of the displacement field and an eigendeformation field that describes the inelastic response of the material beyond the elastic regime. We measure the inelastic contributions of the latter in terms of a non-local energy functional. Our main result shows that, as $\varepsilon \to 0$, the approximate functionals $\Gamma$-converge to a cohesive zone model., Comment: Comments are welcome!

We derive an effective membrane theory in the thin film limit within a two phase material model for a specimen consisting of an elastic matrix and soft inclusions. The soft inclusions may lead to the formation of cracks within the elastic matrix and the corresponding limiting models are described by Griffith type fracture energy functionals. We also provide simplified proofs of relaxation results for bulk materials., Comment: 23 pages. In v2 we correct a mistake in the computation of the $ \Gamma $-limit in the membrane limit. We also make some changes here and there to improve the presentation

We consider a one-dimensional classical many-body system with interaction potential of Lennard-Jones type in the thermodynamic limit at low temperature $1/\beta\in(0,\infty)$. The ground state is a periodic lattice. We show that when the density is strictly smaller than the density of the ground state lattice, the system with $N$ particles fills space by alternating approximately crystalline domains (clusters) with empty domains (voids) due to cracked bonds. The number of domains is of the order of $N\exp(- \beta e_\mathrm{surf}/2)$ with $e_\mathrm{surf}>0$ a surface energy. For the proof, the system is mapped to an effective model, which is a low-density lattice gas of defects. The results require conditions on the interactions between defects. We succeed in verifying these conditions for next-nearest neighbor interactions, applying recently derived uniform estimates of correlations., Comment: 31 pages

We investigate the emergence of rigid polycrystalline structures from atomistic particle systems. The atomic interaction is governed by a suitably normalized pair interaction energy, where the `sticky disk' interaction potential models the atoms as hard spheres that interact when they are tangential. The discrete energy is frame invariant and no underlying reference lattice on the atomistic configurations is assumed. By means of $\Gamma$-convergence, we characterize the asymptotic behavior of configurations with finite surface energy scaling in the infinite particle limit. The effective continuum theory is described in terms of a piecewise constant field delineating the local orientation and micro-translation of the configuration. The limiting energy is local and concentrated on the grain boundaries, i.e., on the boundaries of the zones where the underlying microscopic configuration has constant parameters. The corresponding surface energy density depends on the relative orientation of the two grains, their microscopic translation misfit, and the normal to the interface. We further provide a fine analysis of the surface energies at grain boundaries both for vacuum-solid and solid-solid phase transitions. The latter relies fundamentally on a structure result for grain boundaries showing that due to the extremely brittle setup interpolating boundary layers near cracks are energetically not favorable.

We derive geometrically linearized theories for incompressible materials from nonlinear elasticity theory in the small displacement regime. Our nonlinear stored energy densities may vary on the same (small) length scale as the typical displacements. This allows for applications to multiwell energies as, e.g., encountered in martensitic phases of shape memory alloys and models for nematic elastomers. Under natural assumptions on the asymptotic behavior of such densities we prove Gamma-convergence of the properly rescaled nonlinear energy functionals to the relaxation of an effective model. The resulting limiting theory is geometrically linearized in the sense that it acts on infinitesimal displacements rather than finite deformations, but will in general still have a limiting stored energy density that depends in a nonlinear way on the infinitesimal strains. Our results, in particular, establish a rigorous link of existing finite and infinitesimal theories for incompressible nematic elastomers.

We derive a sharp scaling law for deviations of edge-isoperimetric sets in the lattice $\mathbb Z^d$ from the limiting Wulff shape in arbitrary dimensions. As the number $n$ of elements diverges, we prove that the symmetric difference to the corresponding Wulff set consists of at most $O(n^{(d-1+2^{1-d})/d})$ lattice points and that the exponent $(d-1+2^{1-d})/d$ is optimal. This extends the previously found `$n^{3/4}$ laws' for $d=2,3$ to general dimensions. As a consequence we obtain optimal estimates on the rate of convergence to the limiting Wulff shape as $n$ diverges.

We investigate energetically optimal configurations of thin structures with a pre-strain. Depending on the strength of the pre-strain we consider a whole hierarchy of effective plate theories with a spontaneous curvature term, ranging from linearised Kirchhoff to von K\'arm\'an to linearised von K\'arm\'an theories. While explicit formulae are available in the linearised regimes, the von K\'arm\'an theory turns out to be critical and a phase transition from cylindrical (as in linearised Kirchhoff) to spherical (as in von linearised K\'arm\'an) configurations is observed there. We analyse this behavior with the help of a whole family $(\mathcal{I}^{\theta}_{\rm vK})_{\theta \in (0,\infty)}$ of effective von K\'arm\'an functionals which interpolates between the two linearised regimes. We rigorously show convergence to the respective explicit minimisers in the asymptotic regimes $\theta \to 0$ and $\theta \to \infty$. Numerical experiments are performed for general $\theta \in (0,\infty)$ which indicate a stark transition at a critical value of $\theta$.

We derive von-K\'arm\'an plate theory from three dimensional, purely atomistic models with classical particle interaction. This derivation is established as a $\Gamma$-limit when considering the limit where the interatomic distance $\varepsilon$ as well as the thickness of the plate $h$ tend to zero. In particular, our analysis includes the ultrathin case where $\varepsilon \sim h$, leading to a new von-K\'arm\'an plate theory for finitely many layers.

We derive a hierarchy of plate theories for heterogeneous multilayers from three dimensional nonlinear elasticity by means of $\Gamma$-convergence. We allow for layers composed of different materials whose constitutive assumptions may vary significantly in the small film direction and which also may have a (small) pre-stress. By computing the $\Gamma$-limits in the energy regimes in which the scaling of the pre-stress is non-trivial, we arrive at linearised Kirchhoff, von K{\'a}rm{\'a}n, and fully linear plate theories, respectively, which contain an additional spontaneous curvature tensor. The effective (homogenised) elastic constants of the plates will turn out to be given in terms of the moments of the pointwise elastic constants of the materials.

We analyze the surface energy and boundary layers for a chain of atoms at low temperature for an interaction potential of Lennard-Jones type. The pressure (stress) is assumed small but positive and bounded away from zero, while the temperature $\beta^{-1}$ goes to zero. Our main results are: (1) As $\beta \to \infty$ at fixed positive pressure $p>0$, the Gibbs measures $\mu_\beta$ and $\nu_\beta$ for infinite chains and semi-infinite chains satisfy path large deviations principles. The rate functions are bulk and surface energy functionals $\overline{\mathcal{E}}_{\mathrm{bulk}}$ and $\overline{\mathcal{E}}_\mathrm{surf}$. The minimizer of the surface functional corresponds to zero temperature boundary layers. (2) The surface correction to the Gibbs free energy converges to the zero temperature surface energy, characterized with the help of the minimum of $\overline{\mathcal{E}}_\mathrm{surf}$. (3) The bulk Gibbs measure and Gibbs free energy can be approximated by their Gaussian counterparts. (4) Bounds on the decay of correlations are provided, some of them uniform in $\beta$.

The Adaptive Gain Integrating Pixel Detector (AGIPD) is an x-ray imager, custom designed for the European x-ray Free-Electron Laser (XFEL). It is a fast, low noise integrating detector, with an adaptive gain amplifier per pixel. This has an equivalent noise of less than 1 keV when detecting single photons and, when switched into another gain state, a dynamic range of more than 10$^4$ photons of 12 keV. In burst mode the system is able to store 352 images while running at up to 6.5 MHz, which is compatible with the 4.5 MHz frame rate at the European XFEL. The AGIPD system was installed and commissioned in August 2017, and successfully used for the first experiments at the Single Particles, Clusters and Biomolecules (SPB) experimental station at the European XFEL since September 2017. This paper describes the principal components and performance parameters of the system., Comment: revised version after peer review

We investigate the Edge-Isoperimetric Problem (EIP) for sets with $n$ elements of the cubic lattice by emphasizing its relation with the emergence of the Wulff shape in the crystallization problem. Minimizers $M_n$ of the edge perimeter are shown to deviate from a corresponding cubic Wulff configuration with respect to their symmetric difference by at most $ {\rm O}(n^{3/4})$ elements. The exponent $3/4$ is optimal. This extends to the cubic lattice analogous results that have already been established for the triangular, the hexagonal, and the square lattice in two space dimensions.

We present a variational resolution of the incompressible Navier-Stokes system by means of stabilized Weighted-Inertia-Dissipation-Energy (WIDE) functionals. The minimization of these parameter-dependent functionals corresponds to an elliptic-in-time regularization of the system. By passing to the limit in the regularization parameter along subsequences of WIDE minimizers one recovers a classical Leray-Hopf weak solution.

We prove a homogenization result for Hencky plasticity functionals with non-convex potentials. We also investigate the influence of a small hardening parameter and show that homogenization and taking the vanishing hardening limit commute.

We construct time independent configurations describing a small elastic body moving in a circular orbit in the Schwarzschild spacetime. These configurations are relativistic versions of Newtonian solutions constructed previously by us. In the process we simplify and sharpen previous results of ours concerning elastic bodies in rigid rotation., Comment: 17 pages, second author has chosen to withdraw, small corrections, version identical to the published one

We show existence of solutions for the equations of static atomistic nonlinear elasticity theory on a bounded domain with prescribed boundary values. We also show their convergence to the solutions of continuum nonlinear elasticity theory, with energy density given by the Cauchy-Born rule, as the interatomic distances tend to zero. These results hold for small data close to a stable lattice for general finite range interaction potentials. We also discuss the notion of stability in detail., Comment: new version with only minor changes

We study a planar thin brittle beam subject to elastic deformations and cracks described in terms of a nonlinear Griffith energy functional acting on $SBV$ deformations of the beam. In particular we consider the case in which elastic bulk contributions due to finite bending of the beam are comparable to the surface energy which is necessary to completely break the beam into several large pieces. In the limit of vanishing aspect ratio we rigorously derive an effective Griffith-Euler-Bernoulli functional which acts on piecewise $W^{2,2}$ regular curves representing the midline of the beam. The elastic part of this functional is the classical Euler-Bernoulli functional for thin beams in the bending dominated regime in terms of the curve's curvature. In addition there also emerges a fracture term proportional to the number of discontinuities of the curve and its first derivative.