Partial regularity for local minimizers of functionals with double-phase Orlicz growth

Abstract: In this paper, we consider the local minimizers to a class of non-autonomous functional with double-phase Orlicz growth: \begin{equation*} u \in W^{1,1}(Ω; \mathbb{R}^N) \mapsto \int_Ω\Big( {\varphi_1}\big(|Du|{\mathbb{A}^u}\big) + a(x)\,{\varphi_2} \big(|Du|{\mathbb{A}^u}\big) \Big)\, dx, \end{equation*} where $$|Du|<em>{\mathbb{A}^u} := \big\langle \mathbb{A}(x,u)\,Du, Du \big\rangle<sup>{\frac{1}{2}}$$ with $\mathbb{A}(x,u) = \big{A^{αβ}</em>{ij}(x, u)\big}_{i,j = 1\cdots N}^{α,β= 1\cdots n}$ as the uniformly elliptic bounded symmetric tensor field, and $\varphi_k(\cdot)$ for $k=1,2$ are two different $N$-functions. We prove the partial regularity of their local minimizers if the continuity modulus of $a(\cdot)$ and $\varphi_k(\cdot)$ are fit for the gap conditions, and $\mathbb{A}(x,u)$ meets an optimal regularity in $(x,u)$. This is an extension from the single Orlicz growth functional to the double-phase one based on an essential improvement of several important inequalities.