$ L^p $ Neumann problems in homogenization of general elliptic operators

Li Wang; Qiang Xu; Shulin Zhou

doi:10.3934/dcds.2020210

Article Contents

2020, Volume 40, Issue 8: 5019-5045. Doi: 10.3934/dcds.2020210

This issue Previous Article Blowup results and concentration in focusing Schrödinger-Hartree equation Next Article Asymptotic behavior of sign-changing radial solutions of a semilinear elliptic equation in

$ \mathbb{R}^2 $

when exponent approaches

$ +\infty $

$ L^p $ Neumann problems in homogenization of general elliptic operators

1.
School of Mathematics and Statistics, Lanzhou University, Lanzhou 710000, China
2.
School of Mathematical Sciences, Peking University, Beijing 100871, China

^* Corresponding author: Qiang Xu
^* Corresponding author: Qiang Xu

Received: October 31, 2019

Revised: February 29, 2020

Published: May 2020

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Abstract

In this paper, we extend the nontangential maximal function estimates in $ L^p $-norm obtained by C. Kenig, F. Lin and Z. Shen [13] to the nonhomogeneous elliptic operators with rapidly oscillating periodic coefficients. The result relies on a local Lipschitz boundary estimate, which has not been established in [29]. The present argument develops some new techniques to make the Campanato iteration and real methods workable for general elliptic operators. The result is new even for effective operators, as well as general elliptic equations of scalar.

Keywords:

Homogenization,
Neumann problems,
nontangential maximal function estimates.

Mathematics Subject Classification: Primary: 35J57, 35B27; Secondary: 76M50.

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References

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