Lagrangian mean curvature flow with boundary

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