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In this paper, we first introduce a boundary problem for Lagrangian submanifolds, analogous to the problem for free boundary hypersurfaces and capillary hypersurfaces. Then we present several interesting examples of Lagrangian submanifolds satisfying this boundary condition and we prove a Lagrangian version of the Nitsche (or Hopf) type theorem. Some problems are proposed at the end of this paper.

Lagrangian surfaces with Legendrian boundary