Exploració per autor "Cabré Vilagut, Xavier"
Ara es mostren els items 74-93 de 101
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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 obertWe 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 obertWe 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 obertWe 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 obertWe 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 obertWe 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 obertThe 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 obertThe 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'editorialWe 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'editorialWe 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 obertGiven 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'editorialWe 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 obertWe 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 obertIn 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 obertThese 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 obertWe 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'editorialWe 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 obertWe 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), ...