Warren, MicahBhattacharya, Arunima2019-09-182019-09-182019-09-18https://hdl.handle.net/1794/24840In this thesis, we prove regularity theory for nonlinear fourth order and second order elliptic equations. First, we show that for a certain class of fourth order equations in the double divergence form, where the nonlinearity is in the Hessian, solutions that are $C^{2,\alpha}$ enjoy interior estimates on all derivatives. Next, we consider the fourth order Lagrangian Hamiltonian stationary equation for all phases in dimension two and show that solutions, which are $C^{1,1}$ will be smooth and we also derive a $C^{2,\alpha}$ estimate for it. We also prove explicit $C^{2,\alpha}$ interior estimates for viscosity solutions of fully nonlinear, uniformly elliptic second order equations, which are close to linear equations and we compute an explicit bound for the closeness.en-USAll Rights Reserved.Electronic Thesis or Dissertation