Lipschitz Estimates in Almost‐Periodic Homogenization

By S. Armstrong, Zhongwei Shen. 2014

96 citations recorded by the source corpus.

We establish uniform Lipschitz estimates for second‐order elliptic systems in divergence form with rapidly oscillating, almost‐periodic coefficients. We give interior estimates as well as estimates up to the boundary in bounded C1,α domains with either Dirichlet or Neumann data. The main results extend those in the periodic setting due to Avellaneda and Lin for interior and Dirichlet boundary estimates and later Kenig, Lin, and Shen for the Neumann boundary conditions. In contrast to these papers, our arguments are constructive (and thus the constants are in principle computable) and the results for the Neumann conditions are new even in the periodic setting, since we can treat nonsymmetric coefficients. We also obtain uniform W1,p estimates.© 2016 Wiley Periodicals, Inc.

Subfields: Applied Mathematics, Computational Mechanics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/2afaf9667de9108dc27bd70111098d176220fba6

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In Applied Mathematics

1. S. Armstrong
2. Jean-Christophe Mourrat
3. Tuomo Kuusi
4. Zhongwei Shen
5. Giuseppe Mingione
6. Felix Otto
7. Carlos E. Kenig
8. Giampiero Palatucci

In Computational Mechanics

1. Jean-Christophe Mourrat
2. S. Armstrong
3. Zhongwei Shen
4. Tuomo Kuusi
5. Felix Otto
6. Stefan Neukamm
7. Antoine Gloria

In Computational Theory and Mathematics

1. Jean-Christophe Mourrat
2. S. Armstrong
3. Tuomo Kuusi
4. Zhongwei Shen
5. Felix Otto
6. Antoine Gloria
7. Stefan Neukamm
8. Giuseppe Mingione

Citations in the corpus

• Quantitative Stochastic Homogenization and Large-Scale Regularity (2017, background): “Boundary regularity results appeared previously in [14, 53].”
• Lipschitz Regularity for Elliptic Equations with Random Coefficients (2014, unknown): “[28] C. Kenig, F. Lin, and Z. Shen.”
• Boundary Estimates in Elliptic Homogenization (2015, methodology): “fficients satisfying (1.3), (1.11) and H¨older continuity condition, estimates (6.2) and (6.4) were established in [5, 7, 41, 29]. The results were extended to the case of almost-periodic coefficients in [3]. Also, for Lε with coefficients satisfying (1.2)-(1.3) in Lipschitz domains, some partial results ”
• Mesoscopic Higher Regularity and Subadditivity in Elliptic Homogenization (2015, unknown): “5 in [3] for some similar comments.”