AI 中文总结
本文研究 $H$-系统弱解的正则性，证明在畸变有界时解局部有界，若 $H$ Hölder 连续则解局部 Hölder 连续，方法基于推前测度的 Michael--Simon 型不等式。
AI 中文摘要
我们考虑 $H$-系统 $- \operatorname{div} ( |\mathrm{D} u|^{n-2} \mathrm{D} u ) = (H \circ u) \\, \mathbf{J} u$ 的弱解 $u \in W^{1,n}(U,\mathbf{R}^{n+1})$，其中 $n \ge 2$，$U$ 是 $\mathbf{R}^n$ 中的开单位球，且 $H$ 有界。我们证明，如果 $u$ 的畸变 $|\mathrm{D} u|^n / |\mathbf{J} u|$ 在 $|u|$ 很大的集合上有界，则 $u$ 局部有界。若进一步 $H$ 是 Hölder 连续的，则 $u$ 局部 Hölder 连续。证明通过 $u$ 将 $|\mathrm{D} u|^{n-2} \mathrm{D} u \circ \mathrm{D} u^*$ 推前为 $\mathbf{R}^{n+1}$ 到 $\operatorname{Hom}(\mathbf{R}^{n+1},\mathbf{R}^{n+1})$ 的紧支撑连续映射上的线性泛函，其导数上的值由方程控制。当畸变有界时，该泛函满足 De Philippis、Gennaioli、Pigati 和 Rindler 提出的关于满足一阶偏微分方程的测度的 Michael--Simon 型不等式的假设。该不等式给出 $u$ 推前的 $|\mathrm{D} u|^n \mathscr{L}^n$ 的质量比下界，这些下界蕴含有界性。
英文摘要
We consider weak solutions $u \in W^{1,n}(U,\mathbf{R}^{n+1})$ of the $H$-system $- \operatorname{div} ( |\mathrm{D} u|^{n-2} \mathrm{D} u ) = (H \circ u) \, \mathbf{J} u$, where $n \ge 2$, $U$ is the open unit ball in $\mathbf{R}^n$, and $H$ is bounded. We show that $u$ is locally bounded if its distortion $|\mathrm{D} u|^n / |\mathbf{J} u|$ is bounded on the set where $|u|$ is large. If moreover $H$ is Hölder continuous, then $u$ is locally Hölder continuous. The proof pushes forward $|\mathrm{D} u|^{n-2} \mathrm{D} u \circ \mathrm{D} u^*$ by $u$ to a linear functional on continuous maps of $\mathbf{R}^{n+1}$ into $\operatorname{Hom}(\mathbf{R}^{n+1},\mathbf{R}^{n+1})$ with compact support, whose values on derivatives are controlled by the equation. When the distortion is bounded, this functional satisfies the hypotheses of an inequality of Michael--Simon type for measures satisfying a first-order partial differential equation, due to De Philippis, Gennaioli, Pigati, and Rindler. The inequality yields lower bounds for the mass ratios of the push-forward of $|\mathrm{D} u|^n \mathscr{L}^n$ by $u$, and these imply boundedness.
