Lp solutions of fully nonlinear equations with unbounded lower order terms

The Alexandroff-Backel’man-Pucci maximum principle has turned out to be a powerful and popular tool in the research of elliptic partial differential equations since it was developed in 1960s. Many mathematicians, such as L. Evans and N. Krylov, have made contributions to the regularity and existence theory of fully nonlinear equations by using this maximum principle. Meanwhile, a notion called viscosity solutions, which are a kind of weak solutions that preserve the maximum principle, was developed to deal with equations in non-divergence form. In late 1980s, L. Caffarelli adapted the Alexandroff-Backel’man-Pucci estimates to viscosity solutions and the works on regularity and existence of viscosity solutions by these estimates were started. In this thesis, we prove the existence of viscosity solutions for a variety of fully nonlinear elliptic equations with more general structure conditions. In our setting, the growth rate in the gradient term of the operator is only controlled by an L^p function, whereas, to the best of our knowledge, all the previous results were based on a bounded growth rate. We also prove that in our setting viscosity solutions indeed satisfy the equation almost everywhere. A C^1,1 approximation is constructed in our proof, which solves an open question proposed in the papers of N. Krylov.