We consider mean curvature flow of an initial surface that is the graph a function over some domain definition in $R^n$. If not complete then we impose constant Dirichlet boundary condition at surface. establish longtime-existence and investigate projection flowing onto $R^n$, shadow flow. This moving can be seen as weak solution for hypersurfaces $R^n$ with condition. Furthermore, provide lemm...

We introduce Lagrangian mean curvature flow with boundary in Calabi–Yau manifolds by defining a natural mixed Dirichlet-Neumann condition, and prove that under this flow, the condition is preserved. also study detail of equivariant discs on Lawlor neck self-shrinking Clifford torus, demonstrate long-time existence convergence first instance rescaled second.