Date of this Version
Pieper, M. 2022. Existence and Uniqueness of Minimizers for a Nonlocal Variational Problem. Honors Undergraduate Thesis. University of Nebraska-Lincoln
Nonlocal modeling is a rapidly growing field, with a vast array of applications and connections to questions in pure math. One goal of this work is to present an approachable introduction to the field and an invitation to the reader to explore it more deeply. In particular, we explore connections between nonlocal operators and classical problems in the calculus of variations. Using a well-known approach, known simply as The Direct Method, we establish well-posedness for a class of variational problems involving a nonlocal first-order differential operator. Some simple numerical experiments demonstrate the behavior of these problems for specific choices of kernel and boundary conditions.