双相Orlicz增长泛函局部极小值的部分正则性
Partial regularity for local minimizers of functionals with double-phase Orlicz growth
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- School of Mathematics and Statistics, Beijing Jiaotong University（北京交通大学数学与统计学院）
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中文总结 AI 辅助
本文证明了双相Orlicz增长非自治泛函局部极小值的部分正则性，在连续模间隙条件和最优正则性假设下，将单相结果推广至双相情形。
中文摘要 AI 辅助
本文研究一类具有双相Orlicz增长的非自治泛函的局部极小值：\n\begin{equation*} u \in W^{1,1}(\Omega; \mathbb{R}^N) \mapsto \int_\Omega \Big( {\varphi_1}\big(|Du|_{\mathbb{A}^u}\big) + a(x)\\,{\varphi_2} \big(|Du|_{\mathbb{A}^u}\big) \Big)\\, dx, \end{equation*}\n其中 $|Du|_{\mathbb{A}^u}:= \big\langle \mathbb{A}(x,u)\\,Du, Du \big\rangle^{\frac{1}{2}}$，$\mathbb{A}(x,u) = \big\{A^{\alpha \beta}_{ij}(x, u)\big\}_{i,j = 1\cdots N}^{\alpha,\beta = 1\cdots n}$ 为一致椭圆有界对称张量场，$\varphi_k(\cdot)$（$k=1,2$）为两个不同的 $N$-函数。我们证明，若 $a(\cdot)$ 和 $\varphi_k(\cdot)$ 的连续模满足间隙条件，且 $\mathbb{A}(x,u)$ 在 $(x,u)$ 上满足最优正则性，则其局部极小值具有部分正则性。这是从单个Orlicz增长泛函到双相情形的推广，基于若干重要不等式的基本改进。
英文摘要
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( {φ_1}\big(|Du|_{\mathbb{A}^u}\big) + a(x)\,{φ_2} \big(|Du|_{\mathbb{A}^u}\big) \Big)\, dx, \end{equation*} where $|Du|_{\mathbb{A}^u} := \big\langle \mathbb{A}(x,u)\,Du, Du \big\rangle^{\frac{1}{2}}$ with $\mathbb{A}(x,u) = \big\{A^{αβ}_{ij}(x, u)\big\}_{i,j = 1\cdots N}^{α,β= 1\cdots n}$ as the uniformly elliptic bounded symmetric tensor field, and $φ_k(\cdot)$ 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 $φ_k(\cdot)$ 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.
