Homogenization of a quasilinear elliptic problem with nonlinear Robin boundary conditions
Homogenization of a quasilinear elliptic problem with nonlinear Robin boundary conditions
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DOI：
10.1080/00036811.2011.619982
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发表时间：
2012-05
影响因子：
1.1
通讯作者：
Bituin C. Cabarrubias;P. Donato
中科院分区：
文献类型：
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作者：
Bituin C. Cabarrubias;P. Donato
研究了周期穿孔区域上一类具振荡系数的拟线性椭圆型方程的均匀化问题。在孔的边界上规定了一个非线性Robin条件，该条件依赖于一个真实的参数γ ≥ 1。我们假设数据满足一些适当的假设，这些假设确保，正如Cabarrubias和Donato [B. Cabarrubias和P. Donato，具有非线性Robin条件的拟线性椭圆问题的存在性和唯一性，Carpathian J. Math.（2）（2011）（待出现）]，问题解的存在性和唯一性。特别地，如Cioranescu-Donato-Zaki [D. Cioranescu，P. Donato和R. Zaki，带非线性边界条件的穿孔域椭圆问题的渐近行为，渐近。Anal. 53（2007），pp. 209-235]。在拟线性项上，本文讨论了Chipot [M. Chipot，Elliptic Equations：An Introductory Course，Birkhauser Verlag AG，德国，2009]中描述的并且比Lipschitz条件弱的条件。我们研究收敛到极限问题，这是确定使用周期性开折方法。我们还证明了极限系统的适定性。为了做到这一点，我们证明了均匀化算子继承了初始问题的连续模。因此，齐次拟线性问题的解的唯一性如下。
This article is devoted to the homogenization of a quasilinear elliptic equation with oscillating coefficients in a periodically perforated domain. A nonlinear Robin condition is prescribed on the boundary of the holes, depending on a real parameter γ ≥ 1. We suppose that the data satisfy some suitable hypotheses which ensure, as proved by the authors in Cabarrubias and Donato [B. Cabarrubias and P. Donato, Existence and uniqueness for a quasilinear elliptic problem with nonlinear Robin conditions, Carpathian J. Math. (2) (2011) (to appear)], the existence and the uniqueness of a solution of the problem. In particular, suitable growth conditions are assumed on the nonlinear boundary term, as done in Cioranescu–Donato–Zaki [D. Cioranescu, P. Donato and R. Zaki, Asymptotic behavior of elliptic problems in perforated domains with nonlinear boundary conditions, Asymptot. Anal. 53 (2007), pp. 209–235]. On the quasilinear term, some assumptions on the modulus of continuity introduced in Chipot [M. Chipot, Elliptic Equations: An Introductory Course, Birkhauser Verlag AG, Germany, 2009] and weaker than a Lipschitz condition are prescribed. We study the convergence to a limit problem, which is identified by using the periodic unfolding method. We also prove the well-posedness of the limit system. To do that, we show that the homogenized operator inherits the modulus of continuity of the initial problem. As a consequence, the uniqueness of a solution of the homogenized quasilinear problem follows.