In this talk we discuss the regularity of the Lagrangian Hamiltonian stationary equation which is a fourth order nonlinear PDE. We consider the equation for all phases in dimension two and show that solutions that are $C^{1,1}$ will be smooth. We also derive a $C^{2,α}$ estimate for it.
Note: This talk will be a joint talk of both the geometry/topology seminar and the analysis seminar.