Dissertation Defense: Daniel Maienshein

The dissertation is titled "A Simpler Theorem of Sums for a Class of Non-Uniformly Elliptic PDEs and a Classical Analysis Counterpart of Viterbo's Symplectic Geometry Proof of ABP in the Plane". In the theory of viscosity solutions for second-order, degenerate elliptic PDEs, the theorem of sums is one of the primary analytical tools. Here, we identify a class of PDEs (called Class M) and prove that the theorem of sums takes a simpler form in this class. We present an application of the result by streamlining a comparison principle proof. We explore Class M further by proving existence and uniqueness of viscosity solutions of a certain nonlinear, non-uniformly elliptic PDE. We also provide a classical analysis proof of a version of the Alexandroff-Bakelman-Pucci (ABP) inequality for compactly supported C^2 functions in dimension 2, inspired by the symplectic geometry proof method of Viterbo. We show how the proof may be modified to remove the compact support hypothesis and recover the usual statement of ABP, which includes a boundary term. The various results proven in this thesis are unified by their relation to an open regularity problem in the Heisenberg group. Advisor: Dr. Juan Manfredi