Well-posedness for the mean curvature flow on the half-space and on bounded domains

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 $t\downarrow 0$. 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.