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.
| Identifer | oai:union.ndltd.org:unibo.it/oai:amsdottorato.cib.unibo.it:3073 |
| Date | 05 July 2010 |
| Creators | Cinti, Eleonora <1982> |
| Contributors | Franchi, Bruno |
| Publisher | Alma Mater Studiorum - Università di Bologna |
| Source Sets | Università di Bologna |
| Language | English |
| Detected Language | English |
| Type | Doctoral Thesis, PeerReviewed |
| Format | application/pdf |
| Rights | info:eu-repo/semantics/openAccess |
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