Homogenization of oblique boundary value problems
Homogenization of oblique boundary value problems
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DOI：
10.1515/ans-2022-0051
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发表时间：
2019-11
影响因子：
1.8
通讯作者：
Sunhi Choi;Inwon C. Kim
中科院分区：
文献类型：
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作者：
Sunhi Choi;Inwon C. Kim
摘要考虑一类椭圆算子和边界条件均具有周期振动的非线性Neumann问题。我们的重点是在半空间中提出的问题，但一般正常的方向，可能不平行的方向的周期性。随着振荡频率的增长，定量的均匀化结果被导出。当均匀化算子是旋转不变的，我们证明了Hölder连续的均匀化边界数据。虽然我们遵循Choi和Kim（Homogenization for nonlinear PDE in general domains with oscillating Neumann boundary data，Journal de Mathématiques Pures et Querquées 102（2014），no. 2，419-448）的大纲，但由于我们的问题中边界条件上切向导数的存在，出现了新的挑战。此外，我们还改进和优化了我们方法的收敛速度。我们的结果似乎是新的，即使是线性斜问题。
Abstract We consider a nonlinear Neumann problem, with periodic oscillation in the elliptic operator and on the boundary condition. Our focus is on problems posed in half-spaces, but with general normal directions that may not be parallel to the directions of periodicity. As the frequency of the oscillation grows, quantitative homogenization results are derived. When the homogenized operator is rotation-invariant, we prove the Hölder continuity of the homogenized boundary data. While we follow the outline of Choi and Kim (Homogenization for nonlinear PDEs in general domains with oscillatory Neumann boundary data, Journal de Mathématiques Pures et Appliquées 102 (2014), no. 2, 419–448), new challenges arise due to the presence of tangential derivatives on the boundary condition in our problem. In addition, we improve and optimize the rate of convergence within our approach. Our results appear to be new even for the linear oblique problem.