Stefan Neukamm - WIAS

Stefan Neukamm
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Stefan Neukamm

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Mathematics - Analysis of PDEs (9)
Mathematics - Probability (3)
Mathematics - Numerical Analysis (1)
Mathematics - Functional Analysis (1)

Publications Authored By Stefan Neukamm

We study quantitative periodic homogenization of integral functionals in the context of non-linear elasticity. Under suitable assumptions on the energy densities (in particular frame indifference; minimality, non-degeneracy and smoothness at the identity; $p\geq d$-growth from below; and regularity of the microstructure), we show that in a neighborhood of the set of rotations, the multi-cell homogenization formula of non-convex homogenization reduces to a single-cell formula. The latter can be expressed with help of correctors. Read More

We study the continuum limit of discrete, nonconvex energy functionals defined on crystal lattices in dimensions $d\geq 2$. Since we are interested in energy functionals with random (stationary and ergodic) pair interactions, our problem corresponds to a stochastic homogenization problem. In the non-degenerate case, when the interactions satisfy a uniform $p$-growth condition, the homogenization problem is well-understood. Read More

The qualitative theory of stochastic homogenization of uniformly elliptic linear (but possibly non-symmetric) systems in divergence form is well-understood. Quantitative results on the speed of convergence and on the error in the representative volume method, like those recently obtained by the authors for scalar equations, require a type of stochastic regularity theory for the corrector (e.g. Read More

We establish an optimal, linear rate of convergence for the stochastic homogenization of discrete linear elliptic equations. We consider the model problem of independent and identically distributed coefficients on a discretized unit torus. We show that the difference between the solution to the random problem on the discretized torus and the first two terms of the two-scale asymptotic expansion has the same scaling as in the periodic case. Read More

We consider the corrector equation from the stochastic homogenization of uniformly elliptic finite-difference equations with random, possibly non-symmetric coefficients. Under the assumption that the coefficients are stationary and ergodic in the quantitative form of a Logarithmic Sobolev inequality (LSI), we obtain optimal bounds on the corrector and its gradient in dimensions $d \geq 2$. Similar estimates have recently been obtained in the special case of diagonal coefficients making extensive use of the maximum principle and scalar techniques. Read More

We study the corrector equation in stochastic homogenization for a simplified Bernoulli percolation model on $\mathbb{Z}^d$, $d>2$. The model is obtained from the classical $\{0,1\}$-Bernoulli bond percolation by conditioning all bonds parallel to the first coordinate direction to be open. As a main result we prove (in fact for a slightly more general model) that stationary correctors exist and that all finite moments of the corrector are bounded. Read More

We carry out the spatially periodic homogenization of nonlinear bending theory for plates. The derivation is rigorous in the sense of Gamma-convergence. In contrast to what one naturally would expect, our result shows that the limiting functional is not simply a quadratic functional of the second fundamental form of the deformed plate as it is the case in nonlinear plate theory. Read More

We study the homogenisation of geometrically nonlinear elastic composites with high contrast. The composites we analyse consist of a perforated matrix material, which we call the "stiff" material, and a "soft" material that fills the pores. We assume that the pores are of size $0<\varepsilon\ll 1$ and are periodically distributed with period $\varepsilon$. Read More

We derive, via simultaneous homogenization and dimension reduction, the Gamma-limit for thin elastic plates whose energy density oscillates on a scale that is either comparable to, or much smaller than, the film thickness. We consider the energy scaling that corresponds to Kirchhoff's nonlinear bending theory of plates. Read More

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. Read More