Uniform estimates with data from generalized Lebesgue spaces in periodic structures

Yunsoo Jang

doi:10.1186/s13661-021-01504-x

Abstract

We study various types of uniform Calderón–Zygmund estimates for weak solutions to elliptic equations in periodic homogenization. A global regularity is obtained with respect to the nonhomogeneous term from weighted Lebesgue spaces, Orlicz spaces, and weighted Orlicz spaces, which are generalized Lebesgue spaces, provided that the coefficients have small BMO seminorms and the domains are δ-Reifenberg domains.

Highlights

In this study, our problem is homogenization of elliptic equations of the form ⎧⎨Di(Aij(x)Dju (x)) = Difi(x) in Ω,⎩u (x) = 0 on ∂Ω (1.1)for 1 ≤ i, j ≤ n with n ≥ 2, and 0 < ≤ 1
Our goal is to show uniform Calderón–Zygmund estimates for the weak solutions u with respect to the given nonhomogeneous term f from various spaces, weighted Lebesgue spaces, Orlicz spaces, and weighted Orlicz spaces
Even though our method in this research is applicable to elliptic systems, see [4], and conormal derivative problems, which are a generalization of Neumann problems in nonsmooth domains, see [6], we here consider elliptic equations for simplicity in order to focus on various types of Calderón–Zygmund estimates

Summary

Introduction

For 1 ≤ i, j ≤ n with n ≥ 2, and 0 < ≤ 1. The main difference which arises in the proof is that to obtain uniform estimates in the perturbation method we cannot compare our equation (1.1) to the equation whose coefficient is given by a constant matrix, such as the average value of A, as the proofs in the single equation cases For this reason, different from the single equation case, in the perturbation argument we have to compare (1.1) with (3.6) whose weak solution does not have uniform W 1,∞ regularity. We can derive our desired regularity results from uniform W 1.q (2 < q < ∞) estimates for weak solutions to (3.6), see Lemma 3.3 Through this procedure we apply their own properties of generalized Lebesgue spaces more delicate than single equation cases.

We next let w be a weak solution to

Then we consider that

Therefore p