Lagrangian boundary condition ($u(s,i)\in L_i$)

A Lagrangian boundary condition for a J-holomorphic curve requires specified boundary components of the source to map into specified Lagrangian submanifolds. For the strip, this is $u(s,0)\in L_0$ and $u(s,1)\in L_1$. For a compatible almost complex structure these target submanifolds are totally real. The condition accompanies the interior equation $u_s+J{u}_t=0$.