Browsing by Author "Cabré Vilagut, Xavier"

A mean field equation on a torus: onedimensional symmetry of solutions
Cabré Vilagut, Xavier; Lucia D'Agostino, Marcello; Sanchón Rodellar, Manuel (2003)
Article
Open AccessWe study the equation $$\Delta u=\lambda\left(\frac{e^u}{\int_\oep e^u} \frac{1}{\oep}\right)\quad \text{in }\oep$$ for $u\in E$, where $E = \{ u \in H^1(\oep): u \hbox{ is doubly periodic}, \int_{\oep} u = 0 \}$ and ... 
An extension problem for sums of fractional Laplacians and 1D symmetry of phase transitions
Cabré Vilagut, Xavier; Serra Montolí, Joaquim (20150910)
Article
Open AccessWe study nonlinear elliptic equations for operators corresponding to nonstable Lévy diffusions. We include a sum of fractional Laplacians of different orders. Such operators are infinitesimal generators of nonstable ... 
Antisymmetry of solutions for some weighted elliptic problems
Cabré Vilagut, Xavier; Lucia, Marcello; Sanchón, Manel; Villegas, Salvador (20180317)
Article
Open AccessThis article concerns the antisymmetry, uniqueness, and monotonicity properties of solutions to some elliptic functionals involving weights and a double well potential. In the onedimensional case, we introduce the continuous ... 
A priori estimates for semistable solutions of semilinear elliptic equations
Cabré Vilagut, Xavier; Sanchón Rodellar, Manuel; Spruck, Joel (20160201)
Article
Open AccessWe consider positive semistable solutions u of Lu + f(u) = 0 with zero Dirichlet boundary condition, where L is a uniformly elliptic operator and f is an element of C2 is a positive, nondecreasing, and convex nonlinearity ... 
Boundedness of stable solutions to semilinear elliptic equations: A survey
Cabré Vilagut, Xavier (20170501)
Article
Open AccessThis article is a survey on boundedness results for stable solutions to semilinear elliptic problems. For these solutions, we present the currently known L8 estimates that hold for all nonlinearities. Such estimates are ... 
Curves and surfaces with constant nonlocal mean curvature: Meeting Alexandrov and Delaunay
Cabré Vilagut, Xavier; Fall, Mouhamed Moustapha; SolàMorales Rubió, Joan de (20181201)
Article
Restricted access  publisher's policyWe are concerned with hypersurfaces of RN with constant nonlocal (or fractional) mean curvature. This is the equation associated to critical points of the fractional perimeter under a volume constraint. Our results are ... 
Delaunay hypersurfaces with constant nonlocal mean curvature
Cabré Vilagut, Xavier; Fall, Mouhamed Moustapha; Weth, Tobias (20170812)
Article
Restricted access  publisher's policyWe study hypersurfaces of RN with constant nonlocal (or fractional) mean curvature. This is the equation associated with critical points of the fractional perimeter functional under a volume constraint. We establish the ... 
Energy estimates and 1D symmetry for nonlinear equations involving the halfLaplacian
Cabré Vilagut, Xavier; Cinti, Eleonora (201011)
Article
Restricted access  publisher's policyWe establish sharp energy estimates for some solutions, such as global minimizers, monotone solutions and saddleshaped solutions, of the fractional nonlinear equation 1/2 in R n. Our energy estimates hold for every ... 
Entire solutions of semilinear elliptic equations in R<sup>3</sup> and a conjecture of De Giorgi
Ambrosio, Luigi; Cabré Vilagut, Xavier (1999)
Article
Open Access 
Equacions en derivades parcials, geometria i control estocastic
Cabré Vilagut, Xavier (Societat Catalana de Matemàtiques, 2000)
Article
Open Access 
Existence versus explosion instantanée pour des équations de la chaleur linéaires avec potentiel singulier
Cabré Vilagut, Xavier; Martel, Yvan (1999)
Article
Open AccessIn this Note, we consider the linear heat equation ut  Au= a(x)u in (0,T)xW,u=0 on (0,T)xaW, and u(0)=uº on W, where W C RN is a smooth bounded domain. We assume that a€L^loc(W) a >=0 and u>=. A simple condition on the ... 
Extremal solutions and instantaneous complete blowup for elliptic and parabolic problems
Cabré Vilagut, Xavier (2005)
Article
Open Access 
Front propagation in FisherKPP equations with fractional diffusion
Cabré Vilagut, Xavier; Roquejoffre, JeanMichel (200905)
External research report
Open AccessWe study in this note the FisherKPP equation where the Laplacian is replaced by the generator of a Feller semigroup with slowly decaying kernel, an important example being the fractional Laplacian. Contrary to what happens ... 
Front propagation in FisherKPP equations with fractional diffusion
Cabré Vilagut, Xavier; Roquejoffre, JeanMichel (200912)
Article
Restricted access  publisher's policyFront propagation in Fisher–KPP equations with fractional diffusion.We study in this Note the Fisher–KPP equation where the Laplacian is replaced by the generator of a Feller semigroup with slowly decaying kernel, an ... 
Interior C<sup>2,a</sup> regularity theory for a class of nonconvex fully nonlinear elliptic equations
Cabré Vilagut, Xavier; Caffarelli, Luis A. (2001)
Article
Open AccessWe prove the interior C2,α regularity of solutions for some nonconvex fully nonlinear elliptic equations F(D2u, x) = f (x), x ∈ B1 ⊂ Rn. Our hypothesis is that, for every x ∈ B1, F(·,x) is the minimum of a concave operator ... 
Isoperimetric, Sobolev, and eigenvalue inequalities via the AlexandroffBakelmanPucci method: A survey
Cabré Vilagut, Xavier (20170105)
Article
Open AccessThis paper presents the proof of several inequalities by using the technique introduced by Alexandroff, Bakelman, and Pucci to establish their ABP estimate. First, the author gives a new and simple proof of a lower bound ... 
Layer solutions in a halfspace for boundary reactions
Cabré Vilagut, Xavier; SolàMorales Rubió, Joan de (2005)
Article
Open Access 
Nonlinear equations for fractional laplacians II: existence, uniqueness, and qualitative properties of solutions
Cabré Vilagut, Xavier; Sire, Yannick (20150201)
Part of book or chapter of book
Open AccessThis paper, which is the followup to part I, concerns the equation (Delta)(s)v + G'(v) = 0 in Rn, with s is an element of (0, 1), where (Delta)(s) stands for the fractional Laplacianthe infinitesimal generator of a ... 
Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates
Cabré Vilagut, Xavier; Sire, Yannick (20140101)
Article
Restricted access  publisher's policyThis is the first of two articles dealing with the equation ()sv = f (v) in Rn, with s ¿ (0,1), where ()s stands for the fractional Laplacian — the in¿nitesimal generator of a Lévy process. This equation can be realized ... 
On a longstanding conjecture of E. De Giorgi: old and recent results
Alberti, Giovanni (1965 ); Ambrosio, Luigi; Cabré Vilagut, Xavier (2000)
Article
Open Access