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

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).