Gradient Estimates in the Whole Space for the Double Phase Problems by the Maximal Function Method

IF 0.7 4区数学 Q2 MATHEMATICS

Complex Analysis and Operator Theory Pub Date : 2024-08-01 DOI:10.1007/s11785-024-01579-1

Beilei Zhang, Bin Ge

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引用次数: 0

Abstract

Within this article, the maximal function method is used to establish the Calderón-Zygmund estimates for the weak solutions of a class of non-uniformly elliptic equations

$$\begin{aligned} -\textrm{div}A(x,Du)=-\textrm{div}F(x,f) \quad in \quad {\mathbb {R}}^n, \end{aligned}$$

where $A(x,Du)\approx |Du|^{p_1-2}+\mu (x)|Du|^{p_2-2}$, $F(x,f)\approx |f|^{p_1-2}+\mu (x)|f|^{p_2-2}$ and $1<p_1<p_2$, $0\le \mu (\cdot )\in C^{0,\alpha }({\mathbb {R}}^n),\;\alpha \in (0,1]$. The aforementioned problems arise as Euler-Lagrange equations for variational functionals that were originally presented and studied within the context of Homogenization and the Lavrentiev phenomenon by Marcellini (Arch Ration Mech Anal 105:267–284, 1989. https://doi.org/10.1007/BF00251503) and Zhikov (Izv Akad Nauk SSSR Ser Mat 29:33–66, 1987. https://doi.org/10.1070/IM1987v029n01ABEH000958). They are distinctive in that they exhibit that the growth and ellipticity change between two distinct types of polynomial depending on the position. This feature is characteristic of strongly anisotropic materials. The contribution of this paper is closely tied to the significant advancements made by Colombo and Mingione (J Funct Anal 270:1416–1478, 2016. https://doi.org/10.1016/j.jfa.2015.06.022) in the qualitative analysis of double phase problems, as well as the related techniques used by Zhang et al. (Ann Polon Math, 114:45–65, 2015. https://doi.org/10.4064/ap114-1-4).

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用最大函数法估算双相问题整个空间的梯度

本文使用最大函数法建立了一类非均匀椭圆方程的弱解的 Calderón-Zygmund 估计 $$\begin{aligned} -\textrm{div}A(x、Du)=-\textrm{div}F(x,f) \quad in \quad {\mathbb {R}}^n, \end{aligned}$$ 其中 $A(x,Du)\approx |Du|^{p_1-2}+\mu (x)|Du|^{p_2-2}$, $F(x,f)\approx |f|^{p_1-2}+\mu (x)|f|^{p_2-2}$ and $1<；p_1<p_2$,$0\le \mu (\cdot )\in C^{0,\alpha }({\mathbb {R}}^n),\;\alpha \in (0,1]$.上述问题作为变分函数的欧拉-拉格朗日方程出现，最初是由 Marcellini (Arch Ration Mech Anal 105:267-284, 1989. https://doi.org/10.1007/BF00251503) 和 Zhikov (Izv Akad Nauk SSSR Ser Mat 29:33-66, 1987. https://doi.org/10.1070/IM1987v029n01ABEH000958) 在均质化和拉夫连季耶夫现象的背景下提出并研究的。它们的与众不同之处在于，根据位置的不同，其增长和椭圆度会在两种不同类型的多项式之间发生变化。这一特征是强各向异性材料的特征。本文的贡献与 Colombo 和 Mingione（J Funct Anal 270:1416-1478, 2016. https://doi.org/10.1016/j.jfa.2015.06.022）在双相问题定性分析方面取得的重大进展以及 Zhang 等人（Ann Polon Math, 114:45-65, 2015. https://doi.org/10.4064/ap114-1-4）使用的相关技术密切相关。

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来源期刊

Complex Analysis and Operator Theory 数学-数学

CiteScore

1.20

自引率

12.50%

发文量

107

审稿时长

3 months

期刊介绍： Complex Analysis and Operator Theory (CAOT) is devoted to the publication of current research developments in the closely related fields of complex analysis and operator theory as well as in applications to system theory, harmonic analysis, probability, statistics, learning theory, mathematical physics and other related fields. Articles using the theory of reproducing kernel spaces are in particular welcomed.