  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.
