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} $.
- regularity,
- gradient,
- double phase,
- Calderón-Zygmund estimate,
- non-uniform ellipticity
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

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} $.

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