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- ItemHomogenization of the nonlinear bending theory for plates(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Neukamm, Stefan; Olbermann, HeinerWe carry out the spatially periodic homogenization of Kirchhoff's plate theory. 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 Kirchhoff's plate theory. It turns out that the limiting functional discriminates between whether the deformed plate is locally shaped like a "cylinder" or not. For the derivation we investigate the oscillatory behavior of sequences of second fundamental forms associated with isometric immersions of class W2,2, using two-scale convergence. This is a non-trivial task, since one has to treat two-scale convergence in connection with a nonlinear differential constraint.
- ItemQuantification of ergodicity in stochastic homogenization: Optimal bounds via spectral gap on Glauber dynamics(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Gloria, Antoine; Neukamm, Stefan; Otto, FelixWe study quantitatively the effective large-scale behavior of discrete elliptic equations on the lattice Zd with random coefficients. The theory of stochastic homogenization relates the random, stationary, and ergodic field of coefficients with a deterministic matrix of effective coefficients. This is done via the corrector problem, which can be viewed as a highly degenerate elliptic equation on the infinite-dimensional space of admissible coefficient fields. In this contribution we develop new quantitative methods for the corrector problem based on the assumption that ergodicity holds in the quantitative form of a Spectral Gap Estimate w. r. t. a Glauber dynamics on coefficient fields |as it is the case for independent and identically distributed coefficients. As a main result we prove an optimal decay in time of the semigroup associated with the corrector problem (i. e. of the generator of the process called "random environment as seen from the particle").