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Full-Text Articles in Mathematics
Existence And Uniqueness Of Minimizers For A Nonlocal Variational Problem, Michael Pieper
Existence And Uniqueness Of Minimizers For A Nonlocal Variational Problem, Michael Pieper
Honors Theses
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 …
As Flat As Possible, Jon T. Jacobsen
As Flat As Possible, Jon T. Jacobsen
All HMC Faculty Publications and Research
How does one determine a surface which is as flat as possible, such as those created by soap film surfaces? What does it mean to be as flat as possible? In this paper we address this question from two distinct points of view, one local and one global in nature. Continuing with this theme, we put a temporal twist on the question and ask how to evolve a surface so as to flatten it as efficiently as possible. This elementary discussion provides a platform to introduce a wide range of advanced topics in partial differential equations and helps students …