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Thomas–Fermi Type Variational Problems With Low Regularity / DAMIANO GRECO

Swansea University Author: DAMIANO GRECO

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DOI (Published version): 10.23889/SUThesis.68882

Abstract

This thesis is dedicated to the study of two different Thomas–Fermi type variational problems under optimal assumptions. The first problem concerns studying the existence and qualitative properties ofthe minimizers for a Thomas–Fermi type energy functional with non local repulsion involving a convol...

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Published: Swansea University, Wales, UK 2025
Institution: Swansea University
Degree level: Doctoral
Degree name: Ph.D
Supervisor: Moroz, V., and Finkelshtein, D.
URI: https://cronfa.swan.ac.uk/Record/cronfa68882
first_indexed 2025-02-13T13:09:27Z
last_indexed 2025-02-14T05:46:33Z
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recordtype RisThesis
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spelling 2025-02-13T13:17:42.4122186 v2 68882 2025-02-13 Thomas–Fermi Type Variational Problems With Low Regularity 756c6666c8e58823763c80c15283165e DAMIANO GRECO DAMIANO GRECO true false 2025-02-13 This thesis is dedicated to the study of two different Thomas–Fermi type variational problems under optimal assumptions. The first problem concerns studying the existence and qualitative properties ofthe minimizers for a Thomas–Fermi type energy functional with non local repulsion involving a convolution with the Riesz kernel and an interaction with an external potential. Under mild assumptions, we establish uniqueness and qualitative properties such as positivity, regularity, and decay at infinity of the global minimizer. The second problem concerns the study of optimizers of a Gagliardo–Nirenberg type inequality again involving a convolution with the Riesz kernel. Such a problem is well understood in connection with Keller–Segel models and appears in the study of Thomas–Fermi limit regimes for the Choquard equations with local repulsion and non local attraction. We establish optimal ranges of parameters for the validity of the inequality, discuss the existence and qualitative properties of the optimizers.We further prove that optimizers are either positive, smooth, and fully supported functions or continuous and compactly supported on a ball, or discontinuous and represented as a linear combination of the characteristic function of a ball and a nonconstant nonincreasing Hölder continuous function supported on the same ball. E-Thesis Swansea University, Wales, UK Thomas-Fermi energy, fractional Laplacian, Riesz potential, Asymptotic decay. 15 1 2025 2025-01-15 10.23889/SUThesis.68882 A selection of content is redacted or is partially redacted from this thesis to protect sensitive and personal information. COLLEGE NANME COLLEGE CODE Swansea University Moroz, V., and Finkelshtein, D. Doctoral Ph.D EPSRC Maths DTP 2020 EPSRC Maths DTP 2020 2025-02-13T13:17:42.4122186 2025-02-13T12:58:09.0135087 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics DAMIANO GRECO 1 68882__33580__7b09e52f00c84a30a9a2393582404226.pdf 2024_Greco_D.final.68882.pdf 2025-02-13T13:07:17.1113405 Output 1377898 application/pdf E-Thesis – open access true Copyright: The Author Damiano Greco, 2024 Distributed under the terms of a Creative Commons Attribution 4.0 License (CC BY 4.0). true eng https://creativecommons.org/licenses/by/4.0/
title Thomas–Fermi Type Variational Problems With Low Regularity
spellingShingle Thomas–Fermi Type Variational Problems With Low Regularity
DAMIANO GRECO
title_short Thomas–Fermi Type Variational Problems With Low Regularity
title_full Thomas–Fermi Type Variational Problems With Low Regularity
title_fullStr Thomas–Fermi Type Variational Problems With Low Regularity
title_full_unstemmed Thomas–Fermi Type Variational Problems With Low Regularity
title_sort Thomas–Fermi Type Variational Problems With Low Regularity
author_id_str_mv 756c6666c8e58823763c80c15283165e
author_id_fullname_str_mv 756c6666c8e58823763c80c15283165e_***_DAMIANO GRECO
author DAMIANO GRECO
author2 DAMIANO GRECO
format E-Thesis
publishDate 2025
institution Swansea University
doi_str_mv 10.23889/SUThesis.68882
college_str Faculty of Science and Engineering
hierarchytype
hierarchy_top_id facultyofscienceandengineering
hierarchy_top_title Faculty of Science and Engineering
hierarchy_parent_id facultyofscienceandengineering
hierarchy_parent_title Faculty of Science and Engineering
department_str School of Mathematics and Computer Science - Mathematics{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Mathematics and Computer Science - Mathematics
document_store_str 1
active_str 0
description This thesis is dedicated to the study of two different Thomas–Fermi type variational problems under optimal assumptions. The first problem concerns studying the existence and qualitative properties ofthe minimizers for a Thomas–Fermi type energy functional with non local repulsion involving a convolution with the Riesz kernel and an interaction with an external potential. Under mild assumptions, we establish uniqueness and qualitative properties such as positivity, regularity, and decay at infinity of the global minimizer. The second problem concerns the study of optimizers of a Gagliardo–Nirenberg type inequality again involving a convolution with the Riesz kernel. Such a problem is well understood in connection with Keller–Segel models and appears in the study of Thomas–Fermi limit regimes for the Choquard equations with local repulsion and non local attraction. We establish optimal ranges of parameters for the validity of the inequality, discuss the existence and qualitative properties of the optimizers.We further prove that optimizers are either positive, smooth, and fully supported functions or continuous and compactly supported on a ball, or discontinuous and represented as a linear combination of the characteristic function of a ball and a nonconstant nonincreasing Hölder continuous function supported on the same ball.
published_date 2025-01-15T05:27:51Z
_version_ 1857802020311793664
score 11.096892

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