Wenrui Chang, Yuxia Tong, Shenzhou Zheng
Abstract:
Let \(u \in W^{1,1}(\Omega)\). We study a local gradient continuity of the minimizers
of functionals with borderline double-phase growth,
$$
\mathcal{F}(u ; \Omega) := \int_{\Omega} \Big(|Du|_\mathbb{A}^{p} + a(x)\, |Du|^{p}_\mathbb{A} \log(e + |Du|_\mathbb{A})\Big) \,dx
$$
for \(1 < p < \infty\) and \(|Du|_{\mathbb{A}} :=\langle \mathbb{A}(x) Du, Du \rangle^{1/2}\),
where the matrix \(\mathbb{A}(x) = \{A^{\alpha \beta}(x)\}\) is symmetric with
positive upper and lower bounds of its eigenvalues.
We prove that \(Du\in C^1_{\rm loc}(\Omega)\) based mainly on the
exit-time approach, provided that the moderating factor \(a(x)\) and coefficient
\(\mathbb{A}(x)\) satisfy the so-called Dini-type mean oscillation conditions.
Submitted November 28, 2025. Published April 27, 2026.
Math Subject Classifications: 35B65, 35J15, 35J60.
Key Words: Gradient continuity; non-autonomous functional; Dini mean oscillation; borderline double phase.
DOI: 10.58997/ejde.2026.29
Show me the PDF file (469 KB), TEX file for this article.
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Wenrui Chang School of Mathematics and Statistics Beijing Jiaotong University Beijing 100044, China email: wenruichang@bjtu.edu.cn |
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Yuxia Tong School of Mathematics and Statistics North China University of Science and Technology Tangshan 063210, China email: tongyuxia@126.com |
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Shenzhou Zheng School of Mathematics and Statistics Beijing Jiaotong University Beijing 100044, China email: shzhzheng@bjtu.edu.cn |
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