@article{10549,
abstract = {We derive optimal-order homogenization rates for random nonlinear elliptic PDEs with monotone nonlinearity in the uniformly elliptic case. More precisely, for a random monotone operator on \mathbb {R}^d with stationary law (that is spatially homogeneous statistics) and fast decay of correlations on scales larger than the microscale \varepsilon >0, we establish homogenization error estimates of the order \varepsilon in case d\geqq 3, and of the order \varepsilon |\log \varepsilon |^{1/2} in case d=2. Previous results in nonlinear stochastic homogenization have been limited to a small algebraic rate of convergence \varepsilon ^\delta . We also establish error estimates for the approximation of the homogenized operator by the method of representative volumes of the order (L/\varepsilon )^{-d/2} for a representative volume of size L. Our results also hold in the case of systems for which a (small-scale) C^{1,\alpha } regularity theory is available.},
author = {Fischer, Julian L and Neukamm, Stefan},
issn = {1432-0673},
journal = {Archive for Rational Mechanics and Analysis},
keywords = {Mechanical Engineering, Mathematics (miscellaneous), Analysis},
number = {1},
pages = {343--452},
publisher = {Springer Nature},
title = {{Optimal homogenization rates in stochastic homogenization of nonlinear uniformly elliptic equations and systems}},
doi = {10.1007/s00205-021-01686-9},
volume = {242},
year = {2021},
}
@article{8418,
abstract = {For the Restricted Circular Planar 3 Body Problem, we show that there exists an open set U in phase space of fixed measure, where the set of initial points which lead to collision is O(μ120) dense as μ→0.},
author = {Guardia, Marcel and Kaloshin, Vadim and Zhang, Jianlu},
issn = {0003-9527},
journal = {Archive for Rational Mechanics and Analysis},
keywords = {Mechanical Engineering, Mathematics (miscellaneous), Analysis},
number = {2},
pages = {799--836},
publisher = {Springer Nature},
title = {{Asymptotic density of collision orbits in the Restricted Circular Planar 3 Body Problem}},
doi = {10.1007/s00205-019-01368-7},
volume = {233},
year = {2019},
}
@article{8508,
abstract = {We study generic unfoldings of homoclinic tangencies of two-dimensional area-preserving diffeomorphisms (conservative New house phenomena) and show that they give rise to invariant hyperbolic sets of arbitrarily large Hausdorff dimension. As applications, we discuss the size of the stochastic layer of a standard map and the Hausdorff dimension of invariant hyperbolic sets for certain restricted three-body problems. We avoid involved technical details and only concentrate on the ideas of the proof of the presented results.},
author = {Gorodetski, Anton and Kaloshin, Vadim},
issn = {0081-5438},
journal = {Proceedings of the Steklov Institute of Mathematics},
keywords = {Mathematics (miscellaneous)},
number = {1},
pages = {76--90},
publisher = {Springer Nature},
title = {{Conservative homoclinic bifurcations and some applications}},
doi = {10.1134/s0081543809040063},
volume = {267},
year = {2009},
}