Bistable elliptic equations with fractional diffusion

Cinti, Eleonora (2010) Bistable elliptic equations with fractional diffusion, [Dissertation thesis], Alma Mater Studiorum Università di Bologna. Dottorato di ricerca in Matematica, 22 Ciclo. DOI 10.6092/unibo/amsdottorato/3073.
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Abstract

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.

Abstract
Tipologia del documento
Tesi di dottorato
Autore
Cinti, Eleonora
Supervisore
Co-supervisore
Dottorato di ricerca
Scuola di dottorato
Scienze matematiche, fisiche ed astronomiche
Ciclo
22
Coordinatore
Settore disciplinare
Settore concorsuale
Parole chiave
Fractional Laplacian, sharp energy estimates, one-dimensional symmetry
URN:NBN
DOI
10.6092/unibo/amsdottorato/3073
Data di discussione
5 Luglio 2010
URI

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