Ara es mostren els items 74-93 de 101

    • On a long-standing conjecture of E. De Giorgi: old and recent results 

      Alberti, Giovanni (1965- ); Ambrosio, Luigi; Cabré Vilagut, Xavier (2000)
      Article
      Accés obert
    • On the stability of radial solutions of semilinear elliptic equations in all of R<sup>n</sup> 

      Cabré Vilagut, Xavier; Capella Kort, Antonio (2003)
      Article
      Accés obert
      We establish that every nonconstant bounded radial solution u of −?u = f (u) in all of Rn is unstable if n ? 10. The result applies to every C1 nonlinearity f satisfying a generic nondegeneracy condition. In particular, ...
    • Optimal regularity of stable solutions to nonlinear equations involving the p-Laplacian 

      Cabré Vilagut, Xavier; Miraglio, Pietro; Sanchón Rodellar, Manuel (2020-01-01)
      Article
      Accés obert
      We consider the equation-¿pu=f(u)in a smooth bounded domain ofRn, where¿pis thep-Laplaceoperator. Explicit examples of unbounded stable energy solutions are known ifn=p+4pp-1. Instead, whenn<p+4pp-1, stable solutions have ...
    • Positive solutions of nonlinear problems involving the square root of the Laplacian 

      Cabré Vilagut, Xavier; Tan, Jinggang (2009-05)
      Report de recerca
      Accés obert
      We consider nonlinear elliptic problems involving a nonlocal operator: the square root of the Laplacian in a bounded domain with zero Dirichlet boundary conditions. For positive solutions to problems with power nonlinearities, ...
    • Preface of Llavefest: A broad perspective on finite and infinite dimensional dynamical systems 

      Cabré Vilagut, Xavier; Delshams Valdés, Amadeu; Gidea, Marian; Zeng, Chongchun (American Institute of Mathematical Sciences, 2018-12-01)
      Article
      Accés obert
      We prove that for any non-trivial perturbation depending on any two independent harmonics of a pendulum and a rotor there is global instability. The proof is based on the geometrical method and relies on the concrete ...
    • Recent applications of Nirenberg’s classical ideas 

      Sormani, Christina; Cabré Vilagut, Xavier (2016-02-01)
      Article
      Accés obert
    • Regularity of radial minimizers and extremal solutions of semilinear elliptic equations 

      Cabré Vilagut, Xavier; Capella Kort, Antonio (2005)
      Article
      Accés obert
      We consider a special class of radial solutions of semilinear equations −?u = g(u) in the unit ball of Rn. It is the class of semi-stable solutions, which includes local minimizers, minimal solutions, and extremal ...
    • Regularity of stable solutions to reaction-diffusion elliptic equations 

      Cabré Vilagut, Xavier (European Mathematical Society Press, 2023-07-14)
      Capítol de llibre
      Accés obert
      The boundedness of stable solutions to semilinear (or reaction-diffusion) elliptic PDEs has been studied since the 1970s. In dimensions 10 and higher, there exist stable energy solutions which are unbounded (or singular). ...
    • Regularity of stable solutions to semilinear elliptic equations 

      Cabré Vilagut, Xavier (2019-11-21)
      Audiovisual
      Accés obert
      The regularity of stable solutions to semilinear elliptic PDEs has been studied since the 1970's. In dimensions 10 and higher, there exist singular stable energy solutions. In this talk I will describe a recent work with ...
    • Round table 

      Figalli, Alessio; Vázquez, Juan Luís; Cabré Vilagut, Xavier; Serra, Joaquim; Bonforte, Matteo; Ros Oton, Xavier (2019-11-21)
      Audiovisual
      Accés obert
    • Saddle-shaped solutions of bistable diffusion equations in all of R2m 

      Cabré Vilagut, Xavier; Mourao Terra, Joana (2009)
      Article
      Accés restringit per política de l'editorial
      We study the existence and instability properties of saddle-shaped solutions of the semilinear elliptic equation 􀀀1u D f .u/ in the whole R2m, where f is of bistable type. It is known that in dimension 2m D 2 there ...
    • Sharp energy estimates for nonlinear fractional diffusion equations 

      Cabré Vilagut, Xavier; Cinti, Eleonora (2014-01)
      Article
      Accés restringit per política de l'editorial
      We study the nonlinear fractional equation (−Δ)su=f(u) in Rn, for all fractions 0<s<1 and all nonlinearities f . For every fractional power s∈(0,1) , we obtain sharp energy estimates for bounded global minimizers and for ...
    • Sharp isoperimetric inequalities via the ABP 

      Cabré Vilagut, Xavier; Ros-Oton, Xavier; Serra, Joaquim (2016-01-01)
      Article
      Accés obert
      Given an arbitrary convex cone of Rn, we find a geometric class of homogeneous weights for which balls centered at the origin and intersected with the cone are minimizers of the weighted isoperimetric problem in the convex ...
    • Sobolev and isoperimetric inequalities with monomial weights 

      Cabré Vilagut, Xavier; Ros Oton, Xavier (2013)
      Article
      Accés restringit per política de l'editorial
      We consider the monomial weight |x1|A1⋯|xn|An in Rn, where Ai⩾0 is a real number for each i=1,…,n, and establish Sobolev, isoperimetric, Morrey, and Trudinger inequalities involving this weight. They are the analogue of ...
    • Stable s-minimal cones in R 3 are flat for s ~ 1 

      Cabré Vilagut, Xavier; Cinti, Eleonora; Serra, Joaquim (2019-01-01)
      Article
      Accés obert
      We prove that half spaces are the only stable nonlocal s-minimal cones in R3, for s¿(0,1) sufficiently close to 1. This is the first classification result of stable s-minimal cones in dimension higher than two. Its proof ...
    • Stable solutions to semilinear elliptic equations are smooth up to dimension 9 

      Cabré Vilagut, Xavier; Figalli, Alessio; Ros Oton, Xavier; Serra, Joaquim (2020-01-01)
      Article
      Accés obert
      In this paper we prove the following long-standing conjecture: stable solutions to semi-linear elliptic equations are bounded (and thus smooth) in dimension n¿9. This result, that was only known to be true for n¿4 , is ...
    • Stable solutions to some elliptic problems: minimal cones, the Allen-Cahn equation, and blow-up solutions 

      Cabré Vilagut, Xavier; Poggesi, Giorgio (2018-01-01)
      Article
      Accés obert
      These notes record the lectures for the CIME Summer Course taught by the first author in Cetraro during the week of June 19–23, 2017. The notes contain the proofs of several results on the classification of stable solutions ...
    • The Bernstein technique for integro-differential equations 

      Cabré Vilagut, Xavier; Dipierro, Serena; Valdinoci, Enrico (Springer Nature, 2022-01-25)
      Article
      Accés obert
      We extend the classical Bernstein technique to the setting of integro-differential operators. As a consequence, we provide first and one-sided second derivative estimates for solutions to fractional equations, including ...
    • The influence of fractional diffusion in Fisher-KPP equations 

      Cabré Vilagut, Xavier; Roquejoffre, Jean-Michel (2013)
      Article
      Accés restringit per política de l'editorial
      We study the Fisher-KPP equation where the Laplacian is replaced by the generator of a Feller semigroup with power decaying kernel, an important example being the fractional Laplacian. In contrast with the case of the ...
    • The optimal exponent in the embedding into the Lebesgue spaces for functions with gradient in the Morrey space 

      Cabré Vilagut, Xavier; Charro Caballero, Fernando (Elsevier, 2021-03-26)
      Article
      Accés obert
      We study a natural question that, apparently, has not been well addressed in the literature. Given functions \upsilon with support in the unit ball B1 \subset \mathbb{R}n and with gradient in the Morrey space Mp,\lambda(B1), ...