A non-periodic and two-dimensional example of elliptic homogenization
Jens Persson
本文主要研究一类可能非周期问题的椭圆均匀化。一个非周期和二维的例子进行了研究，我们数值说明了均匀化矩阵。
The focus in this paper is on elliptic homogenization of a certain kind of possibly non-periodic problems. A non-periodic and two-dimensional example is studied, where we numerically illustrate the homogenized matrix.
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Layer Potential Methods for Elliptic Homogenization Problems
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Convergence Rates and Hölder Estimates in Almost-Periodic Homogenization of Elliptic Systems
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Stochastic-Periodic homogenization of Maxwell's equations with nonlinear and periodic conductivity
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Periodic operators over a component domain and homogenization of some class of quasi-linear elliptic problems in two-component domain with interfacial resistance
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Lower Bound of Nodal Sets in Elliptic Homogenization and Functions with Strong Maximum Principle
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We study the homogenization problem for matrix strongly elliptic operators on $L_2(\mathbb R^d)^n$ of the form $\mathcal A^\varepsilon=-\operatorname{div}A(x,x/\varepsilon)\nabla$. The function $A$ is Lipschitz in the first variable and periodic in the second. We do not require that $A^*=A$, so $\mathcal A^\varepsilon$ need not be self-adjoint. In this paper, we provide, for small $\varepsilon$, two terms in the uniform approximation for $(\mathcal A^\varepsilon-μ)^{-1}$ and a first term in the uniform approximation for $\nabla(\mathcal A^\varepsilon-μ)^{-1}$. Primary attention is paid to prov…
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We develop a new real-variable method for weighted $L^p$ estimates. The method is applied to the study of weighted $W^{1, 2}$ estimates in Lipschitz domains for weak solutions of second-order elliptic systems in divergence form with bounded measurable coefficients. It produces a necessary and sufficient condition, which depends on the weight function, for the weighted $W^{1,2}$ estimate to hold in a fixed Lipschitz domain with a given weight. Using this condition, for elliptic systems in Lipschitz domains with rapidly oscillating, periodic and VMO coefficients, we reduce the problem of weighte…
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 $d\geq 3$, respectively of the order $\varepsilon |\log \varepsilon|^{1/2}$ in case $d=2$. Previous results in nonlinear stochastic homogenization have been limited to a sm…
In this paper, for a family of second-order elliptic equations with rapidly oscillating periodic coefficients and rapidly oscillating periodic potentials, we are interested in the $H^1$ convergence rates and the Dirichlet eigenvalues and bounds of the normal derivatives of Dirichlet eigenfunctions. The $H^1$ convergence rates rely on the Dirichlet correctors and the first-order corrector for the oscillating potentials. And the bound results rely on an $O(\varepsilon)$ estimate in $H^1$ for solutions with Dirichlet condition.
We are concerned with the homogenization of second-order linear elliptic equations with random coefficient fields. For symmetric coefficient fields with only short-range correlations, quantified through a logarithmic Sobolev inequality for the ensemble, we prove that when measured in weak spatial norms, the solution to the homogenized equation provides a higher-order approximation of the solution to the equation with oscillating coefficients. In the case of nonsymmetric coefficient fields, we provide a higher-order approximation (in weak spatial norms) of the solution to the equation with osci…
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