Gradient estimates for the double phase problems in the whole space

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} $.

Keywords

• regularity
• gradient
• double phase
• calderón-zygmund estimate
• non-uniform ellipticity