半空间与有界区域上平均曲率流的适定性
Well-posedness for the mean curvature flow on the half-space and on bounded domains
浏览论文内容
中文总结 AI 辅助
本文研究半空间与光滑有界区域上带齐次Dirichlet边界条件的图平均曲率流，证明了标度临界Lipschitz正则性下的局部适定性，初始Lipschitz半范数足够小时解全局存在，有界区域上解指数收敛到平坦图，核心方法为变系数抛物系统的边界Schauder理论。
中文摘要 AI 辅助
我们研究在半空间和光滑有界区域上、满足齐次Dirichlet边界条件的任意余维数图平均曲率流。在标度临界的Lipschitz正则性下，我们证明了可通过满足边界条件的光滑剖面在$W^{1,\infty}$中逼近的初始图的局部适定性。在此类中，若初始Lipschitz半范数足够小，对应解是全局的；在有界区域上，解还会指数收敛到平坦图。正时间正则性由时间加权Hölder估计量化，其加权量在$t\downarrow0$时保持有界。主要分析要素是变系数抛物系统的边界Schauder理论：在半空间上，该理论结合了系数冻结、奇偶延拓以及图系统特有的边界恒等式；在曲面上，局部化与边界平坦化导出各向异性估计，由此递归恢复法向导数。
英文摘要
We study graphical mean curvature flow in arbitrary codimension over the half-space and over smooth bounded domains, subject to homogeneous Dirichlet boundary conditions. At the scaling-critical Lipschitz regularity, we prove local well-posedness for initial graphs that can be approximated in $W^{1,\infty}$ by smooth profiles compatible with the boundary condition. Within this class, if the initial Lipschitz seminorm is sufficiently small, the corresponding solution is global; on a bounded domain, it also converges exponentially to the flat graph. Positive-time regularization is quantified by time-weighted Hölder estimates whose weighted quantities remain bounded as $t\downarrow0$. The main analytic ingredient is a boundary Schauder theory for variable-coefficient parabolic systems. On the half-space, it combines coefficient freezing with parity extensions and boundary identities intrinsic to the graphical system. On curved domains, localization and boundary flattening lead to anisotropic estimates, from which normal derivatives are recovered recursively.
