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THE NONLOCAL NEUMANN PROBLEM

  • Instead of presuming only local interaction, we assume nonlocal interactions. By doing so, mass at a point in space does not only interact with an arbitrarily small neighborhood surrounding it, but it can also interact with mass somewhere far, far away. Thus, mass jumping from one point to another is also a possibility we can consider in our models. So, if we consider a region in space, this region interacts in a local model at most with its closure. While in a nonlocal model this region may interact with the whole space. Therefore, in the formulation of nonlocal boundary value problems the enforcement of boundary conditions on the topological boundary may not suffice. Furthermore, choosing the complement as nonlocal boundary may work for Dirichlet boundary conditions, but in the case of Neumann boundary conditions this may lead to an overfitted model. In this thesis, we introduce a nonlocal boundary and study the well-posedness of a nonlocal Neu- mann problem. We present sufficient assumptions which guarantee the existence of a weak solution. As in a local model our weak formulation is derived from an integration by parts formula. However, we also study a different weak formulation where the nonlocal boundary conditions are incorporated into the nonlocal diffusion-convection operator. After studying the well-posedness of our nonlocal Neumann problem, we consider some applications of this problem. For example, we take a look at a system of coupled Neumann problems and analyze the difference between a local coupled Neumann problems and a nonlocal one. Furthermore, we let our Neumann problem be the state equation of an optimal control problem which we then study. We also add a time component to our Neumann problem and analyze this nonlocal parabolic evolution equation. As mentioned before, in a local model mass at a point in space only interacts with an arbitrarily small neighborhood surrounding it. We analyze what happens if we consider a family of nonlocal models where the interaction shrinks so that, in limit, mass at a point in space only interacts with an arbitrarily small neighborhood surrounding it.

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Author:Michael Vu
URN:urn:nbn:de:hbz:385-1-20013
DOI:https://doi.org/10.25353/ubtr-xxxx-3414-558a
Document Type:Doctoral Thesis
Language:English
Date of completion:2023/04/19
Publishing institution:Universität Trier
Granting institution:Universität Trier, Fachbereich 4
Date of final exam:2023/03/01
Release Date:2023/04/19
GND Keyword:Analysis; Neumann-Problem; Nichtlokalität; Partielle Differentialgleichung
Number of pages:129 Blätter
First page:2
Last page:129
Licence (German):License LogoCC BY: Creative-Commons-Lizenz 4.0 International

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