TY - JOUR
AB - 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.
AU - Fischer, Julian L
AU - Neukamm, Stefan
ID - 10549
IS - 1
JF - Archive for Rational Mechanics and Analysis
KW - Mechanical Engineering
KW - Mathematics (miscellaneous)
KW - Analysis
SN - 0003-9527
TI - Optimal homogenization rates in stochastic homogenization of nonlinear uniformly elliptic equations and systems
VL - 242
ER -
TY - JOUR
AB - 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.
AU - Guardia, Marcel
AU - Kaloshin, Vadim
AU - Zhang, Jianlu
ID - 8418
IS - 2
JF - Archive for Rational Mechanics and Analysis
KW - Mechanical Engineering
KW - Mathematics (miscellaneous)
KW - Analysis
SN - 0003-9527
TI - Asymptotic density of collision orbits in the Restricted Circular Planar 3 Body Problem
VL - 233
ER -
TY - JOUR
AB - 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.
AU - Gorodetski, Anton
AU - Kaloshin, Vadim
ID - 8508
IS - 1
JF - Proceedings of the Steklov Institute of Mathematics
KW - Mathematics (miscellaneous)
SN - 0081-5438
TI - Conservative homoclinic bifurcations and some applications
VL - 267
ER -