We remark on two free-boundary problems for holomorphic curves in nearly-Kähler 6-manifolds. First, we observe that a curve geodesic ball B of the round 6-sphere meets $$\partial B$$ orthogonally must be totally geodesic. Consequently, obtain rigidity results reflection-invariant $$\mathbb {S}^6$$ and associative cones {R}^7$$ . Second, consider with boundary Lagrangian submanifold strict 6-man...

Motivated by a new kind of initial boundary value problem (IBVP) with free arising in wave-structure interaction, we propose here general approach to one-dimensional IBVP as well transmission problems. For strictly hyperbolic 2 × quasilinear systems, derive sharp linear estimates refined dependence on the source term and control traces solution at boundary. These are used obtain results for pro...