arXiv 2005.058792020-05-12math.AP

间断系数$\bm{p(x)}$ -Laplacian偏微分方程解的部分正则性

Partial Regularity of Solutions to $\bm{p(x)}$-Laplacian PDEs with Discontinuous Coefficients

C. S. Goodrich, M. A. Ragusa, A. Scapellato

备注：: Journal of Differential Equations
DOI：: 10.1016/j.jde.2019.11.026

摘要：

对于$Ω\subseteq\mathbb{R}^{n}$的一个开放和有界的区域，我们认为解决方案$u\in W_{\text{loc}}^{1,p(x)}\big(Ω;\mathbb{R}^{N}\big)$，$N>1$，$p(x)$ -拉普拉斯系统\begin{equation} \nabla\cdot\left(a(x)|Du|^{p(x)-2}Du\right)=0\text{, a.e. }x\inΩ,\notag \end{equation}，其中关于系数函数$x\mapsto a(x)$，我们只假设\begin{equation} a\in W^{1,q}(Ω)\cap L^{\infty}(Ω),\notag \end{equation}，其中$q>1$基本上是任意的。这意味着PDE中的系数可以是高度不规则的，尽管如此，我们仍然可以为每个$0<α<1$恢复\begin{equation} u\in\mathscr{C}_{\text{loc}}^{0,α}\big(Ω_0\big),\notag \end{equation}，其中$Ω_0\subseteqΩ$是一组完整的测量。由于我们采用的变分方法，我们的结果适用于更一般的积分泛函\begin{equation} \int_Ωa(x)|Du|^{p(x)}\ dx.\notag \end{equation}的正则性问题

英文摘要：

For $Ω\subseteq\mathbb{R}^{n}$ an open and bounded region we consider solutions $u\in W_{\text{loc}}^{1,p(x)}\big(Ω;\mathbb{R}^{N}\big)$, with $N>1$, of the $p(x)$-Laplacian system \begin{equation} \nabla\cdot\left(a(x)|Du|^{p(x)-2}Du\right)=0\text{, a.e. }x\inΩ,\notag \end{equation} where concerning the coefficient function $x\mapsto a(x)$ we assume only that \begin{equation} a\in W^{1,q}(Ω)\cap L^{\infty}(Ω),\notag \end{equation} where $q>1$ is essentially arbitrary. This implies that the coefficient in the PDE can be highly irregular, and yet in spite of this we still recover that \begin{equation} u\in\mathscr{C}_{\text{loc}}^{0,α}\big(Ω_0\big),\notag \end{equation} for each $0<α<1$, where $Ω_0\subseteqΩ$ is a set of full measure. Due to the variational methodology that we employ, our results apply to the more general question of the regularity of the integral functional \begin{equation} \int_Ωa(x)|Du|^{p(x)}\ dx.\notag \end{equation}

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英文摘要：

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