Gradient estimates for the double phase problems in the whole space

Bei-Lei Zhang; Bin Ge; Bei-Lei Zhang; Bin Ge

doi:10.3934/era.2023372

Electronic Research Archive

2023, Volume 31, Issue 12: 7349-7364. doi: 10.3934/era.2023372

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Gradient estimates for the double phase problems in the whole space

Bei-Lei Zhang ,
Bin Ge ^,

College of Mathematical Sciences, Harbin Engineering University, Harbin 150001, China

Received: 26 September 2023 Revised: 28 October 2023 Accepted: 13 November 2023 Published: 15 November 2023

This paper presents Calderón-Zygmund estimates for the weak solutions of a class of nonuniformly elliptic equations in $\mathbb{R}^n$ , which are obtained through the use of the iteration-covering method. More precisely, a global Calderón-Zygmund type result

$\begin{equation*} |f|^{p_1}+a(x)|f|^{p_2}\in L^s(\mathbb{R}^n) \Rightarrow |Du|^{p_1}+a(x)|Du|^{p_2}\in L^s(\mathbb{R}^n)\quad {\rm for \; any} \; s>1 \end{equation*}$

is established for the weak solutions of

$\begin{equation*} -{\rm div}A(x, Du) = -{\rm div}F(x, f) \quad {\rm in} \; \mathbb{R}^n, \end{equation*}$

which are modeled on

$\begin{equation*} -{\rm div}(|Du|^{p_1-2}Du+a(x)|Du|^{p_2-2}Du) = -{\rm div}(|f|^{p_1-2}f+a(x)|f|^{p_2-2}f), \end{equation*}$

where $0\leq a(\cdot)\in C^{0, \alpha}(\mathbb{R}^n), \; \alpha\in (0, 1]$ and $1 < p_1 < p_2 < p_1+\frac{\alpha p_1}{n}$ .

Keywords:

Citation: Bei-Lei Zhang, Bin Ge. Gradient estimates for the double phase problems in the whole space[J]. Electronic Research Archive, 2023, 31(12): 7349-7364. doi: 10.3934/era.2023372

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Abstract

$\begin{equation*} |f|^{p_1}+a(x)|f|^{p_2}\in L^s(\mathbb{R}^n) \Rightarrow |Du|^{p_1}+a(x)|Du|^{p_2}\in L^s(\mathbb{R}^n)\quad {\rm for \; any} \; s>1 \end{equation*}$

is established for the weak solutions of

$\begin{equation*} -{\rm div}A(x, Du) = -{\rm div}F(x, f) \quad {\rm in} \; \mathbb{R}^n, \end{equation*}$

which are modeled on

$\begin{equation*} -{\rm div}(|Du|^{p_1-2}Du+a(x)|Du|^{p_2-2}Du) = -{\rm div}(|f|^{p_1-2}f+a(x)|f|^{p_2-2}f), \end{equation*}$

where $0\leq a(\cdot)\in C^{0, \alpha}(\mathbb{R}^n), \; \alpha\in (0, 1]$ and $1 < p_1 < p_2 < p_1+\frac{\alpha p_1}{n}$ .

A correction on

Monotonicity and inequalities related to complete elliptic integrals of the second kind

by Fei Wang, Bai-Ni Guo and Feng Qi. AIMS Mathematics, 2020, 5(3): 2732–2742.

DOI: 10.3934/math.2020176

In Acknowledgments section, the Grant number of "Project for Combination of Education and Research Training at Zhejiang Institute of Mechanical and Electrical Engineering" is missing. Here we give the complete information of this fund.

The changes have no material impact on the conclusion of this article. The original manuscript will be updated ^[1]. We apologize for any inconvenience caused to our readers by this change.

Acknowledgments

This work was partially supported by the Foundation of the Department of Education of Zhejiang Province (Grant No. Y201635387), the National Natural Science Foundation of China (Grant No. 11171307), the Visiting Scholar Foundation of Zhejiang Higher Education (Grant No. FX2018093), and the Project for Combination of Education and Research Training at Zhejiang Institute of Mechanical and Electrical Engineering (Grant No. A027120206).

The authors thank anonymous referees for their careful corrections to, helpful suggestions to, and valuable comments on the original version of this manuscript.

Conflict of interest

The authors declare that they have no conflict of interest.

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Gradient estimates for the double phase problems in the whole space

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