发表机构
Indian Institute of Science（印度科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文统一研究Heisenberg群和欧几里得空间上变指数拟线性p(x)-Laplace方程，建立解的一阶导数临界连续性，提出比已知条件更弱的新充分条件，并统一椭圆与次椭圆情形。
AI 中文摘要
我们研究了在Heisenberg群和欧几里得空间中的区域上，变指数拟线性$\mathfrak{p}\left(x\right)$-Laplace型方程的正则性。我们为解的适当一阶导数建立了临界连续性估计。更精确地说，我们证明对于任何弱解$u \in \mathbb{E}W^{1, \mathfrak{p}(\cdot)}\left(\Omega\right)$，满足\begin{align*} \operatorname{div}_{\mathbb{E}} \left( \mathfrak{a}(x) \lvert \nabla_{\mathbb{E}} u \rvert^{\mathfrak{p}\left(x\right)-2} \nabla_{\mathbb{E}} u \right) = f \qquad \text{ in } \Omega, \end{align*}其中$\Omega \subset \mathbb{E}^{n}$，$\mathfrak{a}$是一个一致正的有界标量函数，指数函数$\mathfrak{p}$一致有界远离$1$和$\infty$，只要$f \in L^{\left( Q_{\mathbb{E}}, 1\right)}\left(\Omega\right)$且$\mathfrak{a}, \mathfrak{p}$满足关于其平均振荡可和性的某些条件，则$\nabla_{\mathbb{E}}u$在$\Omega$中连续。这里$\mathbb{E}^{n}$要么是Heisenberg群$\mathbb{H}_{n}$，要么是欧几里得空间$\mathbb{R}^{n}$，而$\operatorname{div}_{\mathbb{E}}$、$\nabla_{\mathbb{E}}$、$Q_{\mathbb{E}}$分别表示相应的散度、梯度和齐次维数。我们使用欧几里得技巧以统一方式处理椭圆和次椭圆情形，我们对$\mathfrak{a}$和$\mathfrak{p}$的条件是新的，并且比所有已知的充分条件更弱，即使在欧几里得情形下也是如此。然而，所有已知的充分条件都蕴含我们的条件，实现了又一次统一。
英文摘要
We study regularity for the variable exponent quasilinear $\mathfrak{p}\left(x\right)$-Laplace type equation on domains in Heisenberg groups and Euclidean spaces. We establish borderline continuity estimates for the appropriate first order derivatives of solutions. More precisely, we show that for any weak solution $u \in \mathbb{E}W^{1, \mathfrak{p}(\cdot)}\left(Ω\right)$ of \begin{align*} \operatorname{div}_{\mathbb{E}} \left( \mathfrak{a}(x) \lvert \nabla_{\mathbb{E}} u \rvert^{\mathfrak{p}\left(x\right)-2} \nabla_{\mathbb{E}} u \right) = f \qquad \text{ in } Ω, \end{align*} where $Ω\subset \mathbb{E}^{n}$, $\mathfrak{a}$ is a uniformly positive bounded scalar function and the exponent function $\mathfrak{p}$ is uniformly bounded away from $1$ and $\infty,$ $\nabla_{\mathbb{E}}u$ is continuous in $Ω$ as soon as $f \in L^{\left( Q_{\mathbb{E}}, 1\right)}\left(Ω\right)$ and $\mathfrak{a}, \mathfrak{p}$ satisfies some conditions regarding the summability of their mean-oscillations. Here $\mathbb{E}^{n}$ is either the Heisenberg group $\mathbb{H}_{n}$ or the Euclidean space $\mathbb{R}^{n}$ and $\operatorname{div}_{\mathbb{E}}$, $\nabla_{\mathbb{E}}$, $Q_{\mathbb{E}}$ stands for the corresponding divergence, gradient and homogeneous dimension, respectively. We treat the elliptic and subelliptic cases in a unified manner using Euclidean techniques and our conditions on $\mathfrak{a}$ and $\mathfrak{p}$ are new and weaker than all the known sufficient conditions even in the Euclidean case. However, all the known sufficient conditions imply our conditions, achieving yet another unification.
