Periodic stationary sets for the fractional perimeter with a general kernel

The aim of this master's thesis is to construct some periodic sets in the plane with constant nonlocal (or fractional) mean curvature induced by general anisotropic kernels. Inspired by the article \cite{bib1} by Cabré, Moustapha, Solà-Morales and Weth, we employ a Lyapunov-Schmidt reduction te...

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Detalhes bibliográficos
Autor: Alcover Borràs, Maties Francesc
Formato: tesis de maestría
Fecha de publicación:2023
País:España
Recursos:Universitat Politècnica de Catalunya (UPC)
Repositorio:UPCommons. Portal del coneixement obert de la UPC
Idioma:inglés
OAI Identifier:oai:upcommons.upc.edu:2117/392899
Acesso em linha:https://hdl.handle.net/2117/392899
Access Level:acceso abierto
Palavra-chave:Differential equations, Partial
Kernel functions
Nonlocal mean curvature
fractional perimeter
nonlocal Delaunay cylinders
fractional Laplacian
minimal surfaces
Equacions en derivades parcials
Kernel, Funcions de
Classificació AMS::35 Partial differential equations
Àrees temàtiques de la UPC::Matemàtiques i estadística
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oai_identifier_str oai:upcommons.upc.edu:2117/392899
network_acronym_str ES
network_name_str España
repository_id_str
spelling Periodic stationary sets for the fractional perimeter with a general kernelAlcover Borràs, Maties FrancescDifferential equations, PartialKernel functionsNonlocal mean curvaturefractional perimeternonlocal Delaunay cylindersfractional Laplacianminimal surfacesEquacions en derivades parcialsKernel, Funcions deClassificació AMS::35 Partial differential equationsÀrees temàtiques de la UPC::Matemàtiques i estadísticaThe aim of this master's thesis is to construct some periodic sets in the plane with constant nonlocal (or fractional) mean curvature induced by general anisotropic kernels. Inspired by the article \cite{bib1} by Cabré, Moustapha, Solà-Morales and Weth, we employ a Lyapunov-Schmidt reduction technique. This approach yields a family of periodic perturbations of an infinite band. To understand the framework and the tools used in the procedure, we introduce fractional Sobolev spaces and the fractional Laplacian. Furthermore, after reviewing the classical theory of minimal surfaces, we extrapolate these concepts into a nonlocal context, obtaining the fractional perimeter and the nonlocal mean curvature. Additionally, we prove the existence of solutions to the Dirichlet problem for the fractional Laplacian.Universitat Politècnica de CatalunyaCabré Vilagut, Xavier20232023-02-0120232023-08-29master thesishttp://purl.org/coar/resource_type/c_bdccNAhttp://purl.org/coar/version/c_be7fb7dd8ff6fe43info:eu-repo/semantics/masterThesisapplication/pdfhttps://hdl.handle.net/2117/392899reponame:UPCommons. Portal del coneixement obert de la UPCinstname:Universitat Politècnica de Catalunya (UPC)Inglésengopen accesshttp://purl.org/coar/access_right/c_abf2info:eu-repo/semantics/openAccessoai:upcommons.upc.edu:2117/3928992026-05-27T15:37:01Z
dc.title.none.fl_str_mv Periodic stationary sets for the fractional perimeter with a general kernel
title Periodic stationary sets for the fractional perimeter with a general kernel
spellingShingle Periodic stationary sets for the fractional perimeter with a general kernel
Alcover Borràs, Maties Francesc
Differential equations, Partial
Kernel functions
Nonlocal mean curvature
fractional perimeter
nonlocal Delaunay cylinders
fractional Laplacian
minimal surfaces
Equacions en derivades parcials
Kernel, Funcions de
Classificació AMS::35 Partial differential equations
Àrees temàtiques de la UPC::Matemàtiques i estadística
title_short Periodic stationary sets for the fractional perimeter with a general kernel
title_full Periodic stationary sets for the fractional perimeter with a general kernel
title_fullStr Periodic stationary sets for the fractional perimeter with a general kernel
title_full_unstemmed Periodic stationary sets for the fractional perimeter with a general kernel
title_sort Periodic stationary sets for the fractional perimeter with a general kernel
dc.creator.none.fl_str_mv Alcover Borràs, Maties Francesc
author Alcover Borràs, Maties Francesc
author_facet Alcover Borràs, Maties Francesc
author_role author
dc.contributor.none.fl_str_mv Cabré Vilagut, Xavier
dc.subject.none.fl_str_mv Differential equations, Partial
Kernel functions
Nonlocal mean curvature
fractional perimeter
nonlocal Delaunay cylinders
fractional Laplacian
minimal surfaces
Equacions en derivades parcials
Kernel, Funcions de
Classificació AMS::35 Partial differential equations
Àrees temàtiques de la UPC::Matemàtiques i estadística
topic Differential equations, Partial
Kernel functions
Nonlocal mean curvature
fractional perimeter
nonlocal Delaunay cylinders
fractional Laplacian
minimal surfaces
Equacions en derivades parcials
Kernel, Funcions de
Classificació AMS::35 Partial differential equations
Àrees temàtiques de la UPC::Matemàtiques i estadística
description The aim of this master's thesis is to construct some periodic sets in the plane with constant nonlocal (or fractional) mean curvature induced by general anisotropic kernels. Inspired by the article \cite{bib1} by Cabré, Moustapha, Solà-Morales and Weth, we employ a Lyapunov-Schmidt reduction technique. This approach yields a family of periodic perturbations of an infinite band. To understand the framework and the tools used in the procedure, we introduce fractional Sobolev spaces and the fractional Laplacian. Furthermore, after reviewing the classical theory of minimal surfaces, we extrapolate these concepts into a nonlocal context, obtaining the fractional perimeter and the nonlocal mean curvature. Additionally, we prove the existence of solutions to the Dirichlet problem for the fractional Laplacian.
publishDate 2023
dc.date.none.fl_str_mv 2023
2023-02-01
2023
2023-08-29
dc.type.none.fl_str_mv master thesis
http://purl.org/coar/resource_type/c_bdcc
NA
http://purl.org/coar/version/c_be7fb7dd8ff6fe43
dc.type.openaire.fl_str_mv info:eu-repo/semantics/masterThesis
format masterThesis
dc.identifier.none.fl_str_mv https://hdl.handle.net/2117/392899
url https://hdl.handle.net/2117/392899
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
dc.rights.openaire.fl_str_mv info:eu-repo/semantics/openAccess
rights_invalid_str_mv open access
http://purl.org/coar/access_right/c_abf2
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv Universitat Politècnica de Catalunya
publisher.none.fl_str_mv Universitat Politècnica de Catalunya
dc.source.none.fl_str_mv reponame:UPCommons. Portal del coneixement obert de la UPC
instname:Universitat Politècnica de Catalunya (UPC)
instname_str Universitat Politècnica de Catalunya (UPC)
reponame_str UPCommons. Portal del coneixement obert de la UPC
collection UPCommons. Portal del coneixement obert de la UPC
repository.name.fl_str_mv
repository.mail.fl_str_mv
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score 15,301603

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