Anja Schlömerkemper

BOUNDARY LAYER ENERGIES FOR NONCONVEX DISCRETE SYSTEMS

In this work we consider a one-dimensional chain of atoms which interact through nearest and next-to-nearest neighbour interactions of Lennard–Jones type. We impose Dirichlet boundary conditions and in addition prescribe the deformation of the second and last but one atoms of the chain. This corresponds to prescribing the slope at the boundary of the discrete setting. We compute the Γ-limits of zero and first order, where the latter leads to the occurrence of boundary layer contributions to the energy. These contributions depend on whether the chain behaves elastically close to the boundary or whether there is a crack. This in turn depends on the given boundary data. We also analyse the location of fracture in dependence on the prescribed discrete slopes.

Discrete-to-continuum limit of magnetic forces

Abstract We derive a formula for the forces within a magnetized body, starting from a discrete configuration of magnetic dipoles on a Bravais lattice. The resulting force consists of the usual (nonlocal) volume term and an additional local surface term, whose coefficients involve a singular sum over the lattice. The force thus obtained is different from the usual continuum expression, reflecting the different character of the lattice regularization of the underlying hypersingular integral. To cite this article: S. Muller, A. Schlomerkemper, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 393–398.

Stochastic Homogenization of Nonconvex Discrete Energies with Degenerate Growth

Recently, there has been considerable effort to understand periodic and stochastic homogenization of elliptic equations and integral functionals with degenerate growth, as well as related questions on the effective behavior of conductance models in degenerate, random environments. In the present paper we prove stochastic homogenization results for nonconvex energy functionals with degenerate growth under moment conditions. In particular, we study the continuum limit of discrete, nonconvex energy functionals defined on crystal lattices in dimensions

About Solutions of Poisson's Equation with Transition Condition in Non-Smooth Domains

d\geq 2

PERTURBATION THEORY OF SCHRODINGER OPERATORS IN INFINITELY MANY COUPLING PARAMETERS

. We consider energy functionals with random (stationary and ergodic) pair interactions; thus our problem corresponds to a stochastic homogenization problem. In the nondegenerate case, when the interactions satisfy a uniform

On a

p

\Gamma

-growth condition, the homogenization problem is well understood. In this paper, we are interested in a degenerate situation, when the interactions satisfy a uniform growth condition neither from above nor neither from below. We...

-Convergence Analysis of a Quasicontinuum Method

Starting from integral representations of solutions of Poisson’s equation with transition condition, we study the first and second derivatives of these solutions for all dimensions d ≥ 2. This involves derivatives of single layer potentials and Newton potentials, which we regularize smoothly. On smooth parts of the boundary of the non-smooth domains under consideration, the convergence of the first derivative of the solution is uniform; this is well known in the literature for regularizations using a sharp cut-off by balls. Close to corners etc. we prove convergence in L1 with respect to the surface measure. Furthermore we show that the second derivative of the solution is in L1 on the bulk. The interface problem studied in this article is obtained from the stationary Maxwell equations in magnetostatics and was initiated by work on magnetic forces. MSC (2000): 35J05, 31A10, 31B10, 78A30

Existence of Weak Solutions to an Evolutionary Model for Magnetoelasticity

In this paper we study the behaviour of Hamilton operators and their spectra which depend on infinitely many coupling parameters or, more generally, parameters taking values in some Banach space. One of the physical models which motivates this framework is a quantum particle moving in a more or less disordered medium. One may, however, also envisage other scenarios where operators are allowed to depend on interaction terms in a manner we are going to discuss below. The central idea is to vary the occurring infinitely many perturbing potentials independently. As a side aspect this then leads naturally to the analysis of a couple of interesting questions of a more or less purely mathematical flavour which belong to the field of infinite-dimensional holomorphy or holomorphy in Banach spaces. In this general setting we study in particular the stability of the self-adjointness of the operators under discussion and the analyticity of eigenvalues under the condition that the perturbing potentials belong to certain classes.

A note on locking materials and gradient polyconvexity

In this paper, we investigate a quasicontinuum method by means of analytical tools. More precisely, we compare a discrete-to-continuum analysis of an atomistic one-dimensional model problem with a corresponding quasicontinuum model. We consider next and next-to-nearest neighbor interactions of Lennard-Jones type and focus on the so-called quasinonlocal quasicontinuum approximation. Our analysis, which applies