Resumen de Bitable elliptic equations with fractional diffusion

Eleonora Cinti

• This work concerns the study of bounded solutions to elliptic nonlinear equations with fractional diffusion of the form ?-?? s u= f ?u? in Rn . More precisely, the aim of this thesis is to investigate some open questions related to the analog of a conjecture of De Giorgi for these equations. The conjecture concerns the 1-D symmetry of bounded monotone solutions in all space, at least up to dimension 8. Of special interest is the bistable elliptic or Allen-Cahn equation, involving fractional Laplacians, which models phase transitions. This property on 1-D symmetry of monotone solutions for the fractional equation was known when n=2 for every fractional power 0?s?1 . The question remained open for n?2 .

  Recently the fractional Laplacians attract much interest in nonlinear analysis. Caffarelli and Silvestre have given a new formulation of the fractional Laplacians through Dirichlet-Neumann maps.

  To study the nonlocal problem ?-?? s u= f ?u? in Rn , we use this formulation which let us to realize it as a local problem in R? n?1 with a nonlinear Neumann condition.

  In this work we focus our attention in two directions. First, in chapter 2, we study a particular type of solutions of ?-?? s u= f ?u? for s=1/2 , which are called saddle-shaped solutions. A crucial property of saddle-shaped solutions is that their 0-level set is the Simons cone. This cone appears in the theory of minimal surfaces and its variational properties motivated the conjecture of De Giorgi, namely the fact that the Simons cone is a minimal cone in dimensions 2m=8 . We are interested in the study of saddle-shaped solutions, because they are the candidates to be global minimizers not 1-D in dimensions 2m=8 (open problem).

  In this first part the main results are: existence of saddle-shaped solutions, as well as their asymptotic behaviour and monotonicity properties in every even dimension 2m, and their instability in dimensions 2m=4 and 2m=6.

  In the second part of this thesis, we give a positive answer to the analog of the conjecture of De Giorgi for the fractional equation with 1/2=s?1 in dimension n=3 . To prove this 1-D symmetry result, we use a Liouville-type argument. In this approach the two principal ingredients in the proof of our 1-D symmetry result are the stability of monotone solutions and a certain energy estimate. In chapters 3 and 4 we establish sharp energy estimates for global minimizers and bounded monotone solutions of our fractional equation for every 0?s?1 and dimension n . As a consequence we deduce the analog of the conjecture of De Giorgi for the fractional equation ?-?? s u= f ?u? , in dimension n=3 for every 1/2=s?1 . To prove our energy estimates we use a comparison argument combined with some extension results for functions belonging to fractional Sobolev spaces.