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Register a new userRegularity of Homogenized Boundary Data in Periodic Homogenization of Elliptic Systems
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This paper is concerned with periodic homogenization of second-order elliptic systems in divergence form with oscillating Dirichlet data or Neumann data of first order. We prove that the homogenized boundary data belong to $W^{1, p}$ for any $1<p<infty$. In particular, this implies that the boundary layer tails are Holder continuous of order $alpha$ for any $alpha in (0,1)$.
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This paper is concerned with the regularity theory of a transmission problem arising in composite materials. We give a new self-contained proof for the $C^{k,alpha}$ estimates on both sides of the interface under the minimal assumptions on the interface and data. Moreover, we prove the uniform Lipschitz estimate across a $C^{1,alpha}$ interface when the coefficients on both sides of the interface are periodic with independent structures and oscillating at different microscopic scales.
We prove explicit doubling inequalities and obtain uniform upper bounds (under $(d-1)$-dimensional Hausdorff measure) of nodal sets of weak solutions for a family of linear elliptic equations with rapidly oscillating periodic coefficients. The doubling inequalities, explicitly depending on the doubling index, are proved at different scales by a combination of convergence rates, a three-ball inequality from certain analyticity, and a monotonicity formula of a frequency function. The upper bounds of nodal sets are shown by using the doubling inequalities, approximations by harmonic functions and an iteration argument.
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 (i.e. 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 $dgeq 3$, respectively 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.
In this paper, we consider a family of second-order elliptic systems subject to a periodically oscillating Robin boundary condition. We establish the qualitative homogenization theorem on any Lipschitz domains satisfying a non-resonance condition. We also use the quantitative estimates of oscillatory integrals to obtain the dimension-dependent convergence rates in $L^2$, assuming that the domain is smooth and strictly convex.
This paper is devoted to establishing the uniform estimates and asymptotic behaviors of the Greens functions $(G_varepsilon,Pi_varepsilon)$ (and fundamental solutions $(Gamma_varepsilon, Q_varepsilon)$) for the Stokes system with periodically oscillating coefficients (including a system of linear incompressible elasticity). Particular emphasis will be placed on the new oscillation estimates for the pressure component $Pi_varepsilon$. Also, for the first time we prove the textit{adjustable} uniform estimates (i.e., Lipschitz estimate for velocity and oscillation estimate for pressure) by making full use of the Greens functions. Via these estimates, we establish the asymptotic expansions of $G_varepsilon, abla G_varepsilon, Pi_varepsilon$ and more, with a tiny loss on the errors. Some estimates obtained in this paper are new even for Stokes system with constant coefficients, and possess potential applications in homogenization of Stokes or elasticity system.
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