Lagrangian caps

We establish an $h$-principle for exact Lagrangian embeddings with concave Legendrian boundary. We prove, in particular, that in the complement of the unit ball $B$ in the standard symplectic ${\mathbb{R}}^{2n},2n\ge 6$, there exists an embedded Lagrangian $n$-disc transversely attached to $B$ along its Legendrian boundary.