Some Graph Laplacians and Variational Methods Applied to Partial Differential Equations on Graphs

In this dissertation we will be examining partial differential equations on graphs. We start by presenting some basic graph theory topics and graph Laplacians with some minor original results. We move on to computing original Jost graph Laplacians of friendly labelings of various finite graphs. We then continue on to a host of original variational problems on a finite graph. The first variational problem is an original basic minimization problem. Next, we use the Lagrange multiplier approach to the Kazdan-Warner equation on a finite graph, our original results generalize those of Dr. Grigor’yan, Dr. Yang, and Dr. Lin. Then we do an original saddle point approach to the Ahmad, Lazer, and Paul resonant problem on a finite graph. Finally, we tackle an original Schrödinger operator variational problem on a locally finite graph inspired by some papers written by Dr. Zhang and Dr. Pankov. The main keys to handling this difficult breakthrough Schrödinger problem on a locally finite graph are Dr. Costa’s definition of uniformly locally finite graph and the locally finite graph analog Dr. Zhang and Dr. Pankov’s compact embedding theorem when a coercive potential function is used in the energy functional. It should also be noted that Dr. Zhang and Dr. Pankov’s deeply insightful Palais-Smale and linking arguments are used to inspire the bulk of our original linking proof