We prove a compactness result for holomorphic curves with boundary on an immersed Lagrangian submanifold with clean self-intersection. As an important consequence, we show that the number of intersections of such holomorphic curves with the self-intersection locus is uniformly bounded in terms of the Hofer energy.
"Compactness for holomorphic curves with switching Lagrangian boundary conditions." J. Symplectic Geom. 8 (3) 267 - 298, September 2010.