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Bistable elliptic equations with fractional diffusion

This work concerns the study of bounded solutions to elliptic
nonlinear equations with fractional diffusion. More precisely, the aim of this
thesis is to investigate some open questions related to a conjecture of De Giorgi about the one-dimensional symmetry of bounded monotone solutions in all
space, at least up to dimension 8.
This property on 1-D symmetry of monotone solutions for
fractional equations was known in dimension n=2. The question remained open for n>2. In this work we establish new sharp energy estimates and one-dimensional symmetry property in dimension 3 for certain solutions of fractional equations. Moreover we study a particular type of solutions, called saddle-shaped solutions, which are the candidates to be global minimizers not one-dimensional in dimensions bigger or equal than 8. This is an open problem and it is expected to be true from the classical theory of minimal surfaces.

Identiferoai:union.ndltd.org:unibo.it/oai:amsdottorato.cib.unibo.it:3073
Date05 July 2010
CreatorsCinti, Eleonora <1982>
ContributorsFranchi, Bruno
PublisherAlma Mater Studiorum - Università di Bologna
Source SetsUniversità di Bologna
LanguageEnglish
Detected LanguageEnglish
TypeDoctoral Thesis, PeerReviewed
Formatapplication/pdf
Rightsinfo:eu-repo/semantics/openAccess

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