The New Jersey Institute of Technology's
Electronic Theses & Dissertations Project

Title:	Eigenvalue problems for fully nonlinear elliptic partial differential equations with transport boundary conditions
Author:	Lesniewski, Jacob
View Online:	njit-etd2021-011 (xiii, 142 pages ~ 5.5 MB pdf)
Department:	Department of Mathematical Sciences
Degree:	Doctor of Philosophy
Program:	Mathematical Sciences
Document Type:	Dissertation
Advisory Committee:	Hamfeldt, Brittany Froese (Committee chair) Shirokoff, David (Committee member) Siegel, Michael (Committee member) Askham, Travis (Committee member) Lewis, Thomas L. (Committee member)
Date:	2021-05
Keywords:	Eigenvalue problems Finite difference methods Fully nonlinear PDEs Numerical analysis Optimal transport Viscosity solutions
Availability:	Unrestricted
Abstract:	Fully nonlinear elliptic partial differential equations (PDEs) arise in a number of applications. From mathematical finance to astrophysics, there is a great deal of interest in solving them. Eigenvalue problems for fully nonlinear PDEs with transport boundary conditions are of particular interest as alternative formulations of PDEs that require data to satisfy a solvability condition, which may not be known explicitly or may be polluted by noisy data. Nevertheless, these have not yet been well-explored in the literature. In this dissertation, a convergence framework for numerically solving eigenvalue problems for fully nonlinear PDEs is introduced. In addition, existing two-dimensional methods for nonlinear equations are extended to handle transport boundary conditions and eigenvalue problems. Finally, new techniques are designed to enable appropriate discretization of a large range of fully nonlinear three-dimensional equations.