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Stefan Müller

Publications and source records attributed to Stefan Müller.

At least 73 records · Page 4Linked to original sources
arXiv2013

Topological contact dynamics I: symplectization and applications of the energy-capacity inequality

We introduce topological contact dynamics of a smooth manifold carrying a cooriented contact structure, generalizing previous work in the case of a symplectic structure [MO07] or a contact form [BS12]. A topological contact isotopy is not generated by a vector field; nevertheless, the group identities, the transformation law, and classical uniqueness results in the smooth case extend to topological contact isotopies and homeomorphisms, giving rise to an extension of smooth contact dynamics to topological dynamics. Our approach is via symplectization of a contact manifold, and our main tools are an energy-capacity inequality we prove for contact diffeomorphisms, combined with techniques from measure theory on oriented manifolds. We establish non-degeneracy of a Hofer-like bi-invariant pseudo-metric on the group of strictly contact diffeomorphisms constructed in [BD06]. The topological automorphism group of the contact structure exhibits rigidity properties analogous to those of symplectic diffeomorphisms, including C^0-rigidity of contact and strictly contact diffeomorphisms.

math.SG↗
arXiv2013

Confined structures of least bending energy

In this paper we study a constrained minimization problem for the Willmore functional. For prescribed surface area we consider smooth embeddings of the sphere into the unit ball. We evaluate the dependence of the the minimal Willmore energy of such surfaces on the prescribed surface area and prove corresponding upper and lower bounds. Interesting features arise when the prescribed surface area just exceeds the surface area of the unit sphere. We show that (almost) minimizing surfaces cannot be a $C^2$-small perturbation of the sphere. Indeed they have to be nonconvex and there is a sharp increase in Willmore energy with a square root rate with respect to the increase in surface area.

math.AP↗
arXiv2013

Almost conical deformations of thin sheets with rotational symmetry

It has been found in numerical experiments that when one removes a sector from an elastic sheet and glues the edges of the sector back together, the resulting configuration is radially symmetric and nearly conical. We make a rigorous analysis of this setting under two simplyfying assumptions: Firstly, we only consider radially symmetric configurations. Secondly, we consider the so-called von-Kármán limit, where the size of the removed region as well as the deformations are small. We choose free boundary conditions for a sheet of infinite size. We show existence of minimizers of the suitably renormalized free energy functional. As a by-product, we obtain a lower bound for the elastic energy that has been conjectured in the related context of d-cones. Moreover, we determine the shape of minimizers at infinity up to exponentially decaying terms.

math.AP↗
arXiv2013

Topological contact dynamics III: uniqueness of the topological Hamiltonian and C^0-rigidity of the geodesic flow

We prove that a topological contact isotopy uniquely defines a topological contact Hamiltonian. Combined with previous results from [MS11], this generalizes the classical one-to-one correspondence between smooth contact isotopies and their generating smooth contact Hamiltonians and conformal factors to the group of topological contact dynamical systems. Applications of this generalized correspondence include C^0-rigidity of smooth contact Hamiltonians, a transformation law for topological contact dynamical systems, and C^0-rigidity of the geodesic flows of Riemannian manifolds.

math.SG↗
arXiv2012

The uniform Korn - Poincaré inequality in thin domains

We study the Korn-Poincaré inequality: \|u\|_{W^{1,2}(S^h)} < C_h \|D(u)\|_{L^2(S^h)}, in domains S^h that are shells of small thickness of order h, around an arbitrary smooth and closed hypersurface S in R^n. By D(u) we denote the symmetric part of the gradient \nabla u, and we assume the tangential boundary conditions: u\vec n^h = 0 on \partial S^h. We prove that C_h remains uniformly bounded as h tends to 0, for vector fields u in any family of cones (with angle <π/2, uniform in h) around the orthogonal complement of extensions of Killing vector fields on S. We also show that this condition is optimal, as in turn every Killing field admits a family of extensions u^h, for which the ratio: \|u^h\|_{W^{1,2}(S^h)} / \|D(u^h)\|_{L^2(S^h)} blows up as h tends to 0, even if the domains S^h are not rotationally symmetric.

math.CA↗
arXiv2012

Generalized mass action systems: Complex balancing equilibria and sign vectors of the stoichiometric and kinetic-order subspaces

Mass action systems capture chemical reaction networks in homogeneous and dilute solutions. We suggest a notion of generalized mass action systems that admits arbitrary nonnegative power-law rate functions and serves as a more realistic model for reaction networks in intracellular environments. In addition to the chemical complexes and the related stoichiometric subspace, we introduce corresponding kinetic complexes, which represent the nonnegative exponents in the rate functions and determine the kinetic-order subspace. We show that several results of Chemical Reaction Network Theory carry over to the case of generalized mass action kinetics. Our main result essentially states that, if the sign vectors of the stoichiometric and the kinetic-order subspace coincide, there exists a unique positive complex balancing equilibrium in every stoichiometric compatibility class. However, in contrast to classical mass action systems, multiple complex balancing equilibria in one stoichiometric compatibility class are possible in general.

math.DS↗
arXiv2012

Conical singularities in thin elastic sheets

When one slightly pushes a thin elastic sheet at its center into a hollow cylinder, the sheet forms (to a high degree of approximation) a developable cone, or "d-cone" for short. Here we investigate one particular aspect of d-cones, namely the scaling of elastic energy with the sheet thickness. Following recent work of Brandman, Kohn and Nguyen we study the Dirichlet problem of finding the configuration of minimal elastic energy when the boundary values are given by an exact d-cone. We improve their result for the energy scaling. In particular, we show that the deviation from the logarithmic energy scaling is bounded by a constant times the double logarithm of the thickness.

math.AP↗
arXiv2012

Topological contact dynamics II: topological automorphisms, contact homeomorphisms, and non-smooth contact dynamical systems

This sequel to our previous paper [MS11b] continues the study of topological contact dynamics and applications to contact dynamics and topological dynamics. We provide further evidence that the topological automorphism groups of a contact structure and a contact form are the appropriate transformation groups of contact dynamical systems. The article includes an examination of the groups of time-one maps of topological contact and strictly contact isotopies, and the construction of a bi-invariant metric on the latter. Moreover, every topological contact or strictly contact dynamical system is arbitrarily close to a continuous contact or strictly contact dynamical system with the same end point. In particular, the above groups of time-one maps are independent of the choice of norm in the definition of the contact distance. On every contact manifold we construct topological contact dynamical systems with time-one maps that fail to be Lipschitz continuous, and smooth contact vector fields whose flows are topologically conjugate but not conjugate by a contact C^1-diffeomorphism.

math.SG↗
arXiv2012

Korn's second inequality and geometric rigidity with mixed growth conditions

Geometric rigidity states that a gradient field which is $L^p$-close to the set of proper rotations is necessarily $L^p$-close to a fixed rotation, and is one key estimate in nonlinear elasticity. In several applications, as for example in the theory of plasticity, energy densities with mixed growth appear. We show here that geometric rigidity holds also in $L^p+L^q$ and in $L^{p,q}$ interpolation spaces. As a first step we prove the corresponding linear inequality, which generalizes Korn's inequality to these spaces.

math.AP↗
arXiv2012

Finite range decomposition for families of gradient Gaussian measures

Let a family of gradient Gaussian vector fields on $ \mathbb{Z}^d $ be given. We show the existence of a uniform finite range decomposition of the corresponding covariance operators, that is, the covariance operator can be written as a sum of covariance operators whose kernels are supported within cubes of diameters $ \sim L^k $. In addition we prove natural regularity for the subcovariance operators and we obtain regularity bounds as we vary within the given family of gradient Gaussian measures.

math-ph↗
arXiv2012

Derivation of a rod theory for biphase materials with dislocations at the interface

Starting from three-dimensional elasticity we derive a rod theory for biphase materials with a prescribed dislocation at the interface. The stored energy density is assumed to be non-negative and to vanish on a set consisting of two copies of SO(3). First, we rigorously justify the assumption of dislocations at the interface. Then, we consider the typical scaling of multiphase materials and we perform an asymptotic study of the rescaled energy, as the diameter of the rod goes to zero, in the framework of $Γ$-convergence.

math.AP↗
arXiv2012

Helicity of vector fields preserving a regular contact form and topologically conjugate smooth dynamical systems

We compute the helicity of a vector field preserving a regular contact form on a closed three-dimensional manifold, and improve results by J.-M. Gambaudo and É. Ghys [GG97] relating the helicity of the suspension of a surface isotopy to the Calabi invariant of the latter. Based on these results, we provide positive answers to two questions posed by V. I. Arnold [Arn86]. In the presence of a regular contact form that is also preserved, the helicity extends to an invariant of an isotopy of volume preserving homeomorphisms, and is invariant under conjugation by volume preserving homeomorphisms. A similar statement also holds for suspensions of surface isotopies and surface diffeomorphisms. This requires the techniques of topological Hamiltonian and contact dynamics developed in [MO07, Mül08b, Vit06, BS11b, BS11a, MS11]. Moreover, we generalize an example of H. Furstenberg [Fur61] of topologically conjugate but not C^1-conjugate area preserving diffeomorphisms of the two-torus to trivial T^2-bundles, and construct examples of Hamiltonian and contact vector fields that are topologically conjugate but not C^1-conjugate. Higher-dimensional helicities are considered briefly at the end of the paper.

math.SG↗
arXiv2011

On properly essential classical conformal diffeomorphism groups

We prove that various classical conformal diffeomorphism groups, which are known to be essential [1], are in fact properly essential. This is a consequence of a local criterion on a conformal diffeomorphism in the form of a cohomological equation. Furthermore, we study the orbit of a tensor field under the action of the conformal diffeomorphism group for these classical conformal structures. On every closed contact manifold, we find conformal contact forms that are not diffeomorphic.

math.SG↗
arXiv2011

A note on the volume flux of smooth and continuous strictly contact isotopies

This note on the flux homomorphism for strictly contact isotopies complements the recent paper [MS11a] by P. Spaeth and the author. We determine the volume flux restricted to symplectic and volume-preserving contact isotopies and their C^0-limits for some classes of symplectic and contact manifolds and for a number of examples. In particular, we see that the restriction of the flux may fail to be surjective. It vanishes for an isotopy preserving a regular contact form, but can be nontrivial for non-regular contact forms. Applications are discussed in the article cited above. We also find obstructions to regularizing a strictly contact isotopy that are not present for Hamiltonian isotopies [Pol01, Section 5.2] or contact isotopies [MS11b].

math.SG↗
arXiv2011

On the commutability of homogenization and linearization in finite elasticity

We study non-convex elastic energy functionals associated to (spatially) periodic, frame indifferent energy densities with a single non-degenerate energy well at SO(n). Under the assumption that the energy density admits a quadratic Taylor expansion at identity, we prove that the Gamma-limits associated to homogenization and linearization commute. Moreover, we show that the homogenized energy density, which is determined by a multi-cell homogenization formula, has a quadratic Taylor expansion with a quadratic term that is given by the homogenization of the quadratic term associated to the linearization of the initial energy density.

math.AP↗
arXiv2010

Singular kernels, multiscale decomposition of microstructure, and dislocation models

We consider a model for dislocations in crystals introduced by Koslowski, Cuitiño and Ortiz, which includes elastic interactions via a singular kernel behaving as the $H^{1/2}$ norm of the slip. We obtain a sharp-interface limit of the model within the framework of $Γ$-convergence. From an analytical point of view, our functional is a vector-valued generalization of the one studied by Alberti, Bouchitté and Seppecher to which their rearrangement argument no longer applies. Instead we show that the microstructure must be approximately one-dimensional on most length scales and exploit this property to derive a sharp lower bound.

math.AP↗