论文标题
$ \ bm {p（x）} $ - laplacian PDE的部分规律性与不连续系数
Partial Regularity of Solutions to $\bm{p(x)}$-Laplacian PDEs with Discontinuous Coefficients
论文作者
论文摘要
对于$ω\ subseteq \ mathbb {r}^{n} $一个开放和有限的区域，我们考虑解决方案$ u \ in w _ {\ text {loc}}}^{1，p（x）} \ big（ω; \ mathbb {r}系统\ begin {equination} \ nabla \ cdot \ left（a（x）| du |^{p（x）-2} du \ right）= 0 \ text {，a.e. } x \inΩ，\ notag \ end {qore}，其中有关系数函数$ x \ mapsto a（x）$，我们仅假设\ begin {equication} a \ in w^{1，q}（q}（ω）（ω）\ cap l^{\ cap l^{\ infty}（\ infty}（\ cap l^{\ formany），\ nestiral unionly nestance nestiral nesequanty。这意味着PDE中的系数可能高度不规则，但是尽管如此，我们仍然恢复过\ begin {equination} u \ in \ mathscr {c} _ {c} _ {\ text {loc}}}}}^{0，α} \ big（ $ω_0\subseteqΩ$是一组完整度量。由于我们采用的变异方法，我们的结果适用于积分函数\ begin {qore} \int_Ωa（x）| du |^{p（x）} \ dx。
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}
