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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
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 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.
Item Description: A selection of content is redacted or is partially redacted from this thesis to protect sensitive and personal information.
Keywords: Thomas-Fermi energy, fractional Laplacian, Riesz potential, Asymptotic decay.
College: Faculty of Science and Engineering
Funders: EPSRC Maths DTP 2020

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