On Kernels and Antiderivatives of Nonlocal Derivatives

Authors

Alex John Heitzman, University of Nebraska-LincolnFollow

First Advisor

Mikil Foss

Degree Name

Doctor of Philosophy (Ph.D.)

Committee Members

Christopher Schafhauser, Florin Bobaru, Petronela Radu

Department

Mathematics

Date of this Version

8-2025

Document Type

Dissertation

Citation

A dissertation presented to the Graduate College of the University of Nebraska in partial fulfillment of requirements for the degree of Doctor of Philosophy

Major: Mathematics

Under the supervision of Mikil Foss

Lincoln, Nebraska, August 2025

Comments

Copyright 2025, Alex John Heitzman. Used by permission

Abstract

Nonlocal operators are mathematical operators taking functions to other functions f → Df , where to evaluate the operator Df at a point x, one must know the value of f in some region around x, and that region cannot be arbitrarily small. Nonlocal derivatives are like derivatives in that they measure the deviation of a function f(z) from f(x) when z is close to x. In this thesis, we will study nonlocal operators of the form

D_kf(x) = [integral]_Ω [f(x) - f(z)]k(x,z)dz

In the first part of the thesis, we will discuss solutions to equations of the form D_k(f) = g (i.e. “nonlocal antiderivatives” of a function g). We will prove existence and uniqueness, and show how to approximate said solutions. In the second, we will present results about the kernels of nonlocal operators and maximum principles for solutions to D_kf ≤ 0, and show that under various circumstances, the only functions with D_kf = 0 are constant.

Advisor: Mikil Foss

Recommended Citation

Heitzman, Alex John, "On Kernels and Antiderivatives of Nonlocal Derivatives" (2025). Dissertations and Doctoral Documents from University of Nebraska-Lincoln, 2023–. 365.
https://digitalcommons.unl.edu/dissunl/365