Browsing by Author "Cabré Vilagut, Xavier"
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A fractional Michael–Simon Sobolev inequality on convex hypersurfaces
Cabré Vilagut, Xavier; Cozzi, Matteo; Csató, Gyula (Elsevier, 20230101)
Part of book or chapter of book / Nautical chart
Open AccessThe classical Michael–Simon and Allard inequality is a Sobolev inequality for functions defined on a submanifold of Euclidean space. It is governed by a universal constant independent of the manifold, thanks to an additional ... 
A gradient estimate for nonlocal minimal graphs
Cabré Vilagut, Xavier; Cozzi, Matteo (20190401)
Article
Open AccessWe consider the class of measurable functions defined in all of Rn that give rise to a nonlocal minimal graph over a ball of Rn. We establish that the gradient of any such function is bounded in the interior of the ball ... 
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 ... 
A new proof of the boundedness results for stable solutions to semilinear elliptic equations
Cabré Vilagut, Xavier (American Institute of Mathematical Sciences, 20191201)
Article
Open AccessWe consider the class of stable solutions to semilinear equations ¿u=f(u) in a bounded smooth domain of Rn. Since 2010 an interior a priori L8 bound for stable solutions is known to hold in dimensions n=4 for all C1 ... 
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 ... 
A universal Hölder estimate up to dimension 4 for stable solutions to halfLaplacian semilinear equations
Cabré Vilagut, Xavier; Sanz Perela, Tomás (20220425)
Article
Open AccessWe study stable solutions to the equation , posed in a bounded domain of . For nonnegative convex nonlinearities, we prove that stable solutions are smooth in dimensions . This result, which was known only for , follows ... 
A Weierstrass extremal field theory for the fractional Laplacian
Cabré Vilagut, Xavier; Urtiaga Erneta, Iñigo; Felipe Navarro, Juan Carlos (De Gruyter, 20231027)
Article
Restricted access  publisher's policyIn this paper, we extend, for the first time, part of the Weierstrass extremal field theory in the Calculus of Variations to a nonlocal framework. Our model case is the energy functional for the fractional Laplacian (the ... 
ADVANCED COURSE IN PARTIAL DIFFERENTIAL EQUATIONS
Cabré Vilagut, Xavier (Universitat Politècnica de Catalunya, 20170530)
Exam
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ADVANCED COURSE IN PARTIAL DIFFERENTIAL EQUATIONS
Cabré Vilagut, Xavier (Universitat Politècnica de Catalunya, 20160418)
Exam
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ADVANCED COURSE IN PARTIAL DIFFERENTIAL EQUATIONS
Cabré Vilagut, Xavier (Universitat Politècnica de Catalunya, 20160607)
Exam
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ADVANCED COURSE IN PARTIAL DIFFERENTIAL EQUATIONS
Cabré Vilagut, Xavier (Universitat Politècnica de Catalunya, 20170419)
Exam
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ADVANCED COURSE IN PARTIAL DIFFERENTIAL EQUATIONS
Cabré Vilagut, Xavier (Universitat Politècnica de Catalunya, 20180321)
Exam
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ADVANCED COURSE IN PARTIAL DIFFERENTIAL EQUATIONS
Cabré Vilagut, Xavier (Universitat Politècnica de Catalunya, 20180529)
Exam
Restricted access to the UPC academic community 
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 ... 
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 ... 
Calibrations and nullLagrangians for nonlocal perimeters and an application to the viscosity theory
Cabré Vilagut, Xavier (20200204)
Article
Restricted access  confidentiality agreementFor nonnegative even kernels K, we consider the Knonlocal perimeter functional acting on sets. Assuming the existence of a foliation of space made of solutions of the associated Knonlocal mean curvature equation in an ... 
Curves and surfaces with constant nonlocal mean curvature: Meeting Alexandrov and Delaunay
Cabré Vilagut, Xavier; Fall, Mouhamed Moustapha; SolàMorales Rubió, Joan de (20181201)
Article
Open AccessWe 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 ... 
Discurs de l'honoris causa i elogi dels mèrits del professor Alessio Figalli, pel professor Xavier Cabré. 2019
Cabré Vilagut, Xavier (20191122)
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Open Access