Articles and Preprints

• A fractional Michael-Simon Sobolev inequality on convex hypersurfaces, with M. Cozzi and G. Csató. To appear in Ann. I.H.Poincaré
• Universal Hardy-Sobolev inequalities on hypersurfaces of Euclidean space, with P. Miraglio. Commun. Contemp. Math. (2021) 2150063 (25 pages), DOI: 10.1142/S0219199721500632
• The optimal exponent in the embedding into the Lebesgue spaces for functions with gradient in the Morrey space, with F. Charro. Adv. Math. 380 (2021), article number: 107592
• Optimal regularity of stable solutions to nonlinear equations involving the p-Laplacian, with P. Miraglio and M. Sanchon. Adv. Calc. Var. 2020, https://doi.org/10.1515/acv- 2020-0055
• Stable solutions to semilinear elliptic equations are smooth up to dimension 9, with A. Figalli, X. Ros-Oton, and J. Serra. Acta Mathematica 224 (2020), 187–252
• Calibrations and null-Lagrangians for nonlocal perimeters and an application to the viscosity theory. Ann. Mat. Pura Appl. 199 (2020), 1979-1995
• Stable s-minimal cones in ℝ³ are flat for s∼1, with E. Cinti and J. Serra. J. Reine Angew. Math. 764 (2020), 157-180
• A gradient estimate for nonlocal minimal graphs, with M. Cozzi. Duke Math. J. 168 (2019), 775-848
• A new proof of the boundedness results for stable solutions to semilinear elliptic equations. Discrete Contin. Dyn. Syst. 39 (2019), 7249-7264
• Stable solutions to some elliptic problems: minimal cones, the Allen-Cahn equation, and blow-up solutions, with G. Poggesi. Geometry of PDEs and related problems, 1-45, Lecture Notes in Math., 2220, Fond. CIME/CIME Found. Subser., Springer, Cham, 2018
• Antisymmetry of solutions for some weighted elliptic problems, with M. Lucia, M. Sanchon, and S. Villegas. Comm. in Partial Diff. Eq. 43 (2018), 506-547
• Near-sphere lattices with constant nonlocal mean curvature, with M.M. Fall and T. Weth. Mathematische Annalen 370 (2018), 1513-1569
• Delaunay hypersurfaces with constant nonlocal mean curvature, with M.M. Fall and T. Weth. J. Math. Pures Appl. 110 (2018), 32-70​
• Curves and surfaces with constant nonlocal mean curvature: meeting Alexandrov and Delaunay, with M. M. Fall, J. Solà-Morales, and T. Weth.  J. Reine Angew. Math. 745 (2018), 253-280
• Boundedness of stable solutions to semilinear elliptic equations: a survey. Adv. Nonlinear Stud. 17 (2017), 355-368
• Isoperimetric, Sobolev, and eigenvalue inequalities via the ABP method: a survey. Chin. Ann. Math. 38 B (2017), 201-214
• Sharp isoperimetric inequalities via the ABP method, with X. Ros-Oton and J. Serra. Journal of the European Mathematical Society 18 (2016), 2971-2998
• An extension problem for sums of fractional Laplacians and 1-D symmetry of phase transitions, with J. Serra. Nonlinear Analysis-Theory Methods & Applications 137 (2016), 246-265
• A priori estimates for semistable solutions of semilinear elliptic equations, with M. Sanchon and J. Spruck. Discrete and Continuous Dynamical Systems 36 (2016), 601-609
• Traveling wave solutions in a half-space for boundary reactions, with N. Consul and J.V. Mandé. Analysis & PDE 8 (2015), 333-364
• Nonlinear equations for fractional Laplacians II: existence, uniqueness, and qualitative properties of solutions, with Y. Sire. Transactions of the American Mathematical Society 367 (2015), 911-941
• Nonlinear equations for fractional Laplacians I: Regularity, maximum principles, and Hamiltonian estimates, with Y. Sire. Ann. I.H.Poincaré 31 (2014), 23-53
• Sharp energy estimates for nonlinear fractional diffusion equations, with E. Cinti. Calc. Var. and PDEs 49 (2014), 233-269
• The influence of fractional diffusion in Fisher-KPP equations, with J.-M. Roquejoffre. Communications in Mathematical Physics 320 (2013), 679-722
• Regularity of stable solutions up to dimension 7 in domains of double revolution, with X. Ros-Oton. Comm. in Partial Differential Equations 38 (2013), 135-154
• Geometric-type Sobolev inequalities and applications to the regularity of minimizers, with M. Sanchón. Journal of Functional Analysis 264 (2013), 303-325
• Sobolev and isoperimetric inequalities with monomial weights, with X. Ros-Oton. J. Differential Equations 255 (2013), 4312

• Euclidean balls solve some isoperimetric problems with nonradial weights, with X. Ros-Oton and J. Serra. C. R. Acad. Sci. Paris, Ser. I 350 (2012), 945-947
• Propagation in Fisher-KPP type equations with fractional diffusion in periodic media, with A.-C. Coulon and J.-M. Roquejoffre. C. R. Acad. Sci. Paris, Ser. I 350 (2012), 885-890
• Uniqueness and stability of saddle-shaped solutions to the Allen-Cahn equation. Journal de Mathématiques Pures et Appliquées 98 (2012), 239-256
• Energy estimates and 1-D symmetry for nonlinear equations involving the half-Laplacian, with E. Cinti. Discrete and Continuous Dynamical Systems 28 (2010), 1179-1206. A special issue Dedicated to Louis Nirenberg on the Occasion of his 85th Birthday
• Regularity of minimizers of semilinear elliptic problems up to dimension four. Comm. Pure Applied Mathematics 63 (2010), 1362-1380
• Qualitative properties of saddle-shaped solutions to bistable diffusion equations, with J. Terra. Comm. in Partial Differential Equations 35 (2010), 1923-1957
• Positive solutions of nonlinear problems involving the square root of the Laplacian, with Jinggang Tan. Advances in Mathematics 224 (2010), 2052-2093
• Front propagation in Fisher-KPP equations with fractional diffusion, with J.-M. Roquejoffre. C. R. Math. Acad. Sci. Paris 347 (2009), 1361-1366
• Saddle-shaped solutions of bistable diffusion equations in all of R^{2m}, with J. Terra. Jour. of the European Math. Society 11 (2009), 819-843
• Regularity of radial minimizers of reaction equations involving the p-Laplacian, with A. Capella and M. Sanchón. Calculus of Var. and Partial Diff. Eq. 34 (2009), 475-494
• Elliptic PDEs in Probability and Geometry. Symmetry and regularity of solutions. Discrete Contin. Dyn. Syst. 20 (2008), 425-457
• Semi-stable and extremal solutions of reaction equations involving the p-Laplacian, with M. Sanchón. Comm. Pure Appl. Anal. 6 (2007), 43-67
• Regularity of minimizers for three elliptic problems: minimal cones, harmonic maps, and semilinear equations, with A. Capella. Pure and Applied Math Quarterly 3 (2007), 801-825
• Regularity of radial extremal solutions of semilinear elliptic equations. Bol. Soc. Esp. Mat. Apl. 34 (2006), 92-98
• Regularity of radial minimizers and extremal solutions of semilinear elliptic equations, with A. Capella. J. Functional Analysis 238 (2006), 709-733
• Extremal solutions and instantaneous complete blow-up for elliptic and parabolic problems. Contemporary Mathematics, American Math. Soc. 2007, in: Perspectives in Nonlinear Partial Differential Equations: In honor of Haim Brezis
• The parameterization method for invariant manifolds III: overview and applications, with E. Fontich and R. de la Llave. J. Differential Equations 218 (2005), 444-515
• Layer solutions in a half-space for boundary reactions, with J. Solà-Morales. Comm. Pure Appl. Math. 58 (2005), 1678-1732
• A mean field equation on a torus: one-dimensional symmetry of solutions, with M. Lucia and M. Sanchón. Comm. Partial Differential Equations 30 (2005), 1315-1330
• On the stability of radial solutions of semilinear elliptic equations in all of R^n, with A. Capella. C. R. Acad. Sci. Paris, Ser. I 338 (2004), 769-774
• Interior C^{2,\alpha} regularity theory for a class of nonconvex fully nonlinear elliptic equations, with L. Caffarelli. J. Math. Pures Appl. 82 (2003), 573-612
• The parameterization method for invariant manifolds II: regularity with respect to parameters, with E. Fontich and R. de la Llave. Indiana Univ. Math. J. 52 (2003), 329-360
• The parameterization method for invariant manifolds I: manifolds associated to non-resonant subspaces, with E. Fontich and R. de la Llave. Indiana Univ. Math. J. 52 (2003), 283-328
• Topics in regularity and qualitative properties of solutions of nonlinear elliptic equations. Discrete and Continuous Dynamical Systems 8 (2002), 331-359
• A conjecture of De Giorgi on symmetry for elliptic equations in R^n. European Congress of Math. Progr. Math. I, vol. 201, 259-265. Birkhauser, 2001
• On a long-standing conjecture of E. De Giorgi: symmetry in 3D for general nonlinearities and a local minimality property, with G. Alberti and L. Ambrosio. Acta Applicandae Mathematicae 65 (2001), 9-33
• Entire solutions of semilinear elliptic equations in R^3 and a conjecture of De Giorgi, with L. Ambrosio. J. Amer. Math. Soc. 13 (2000), 725-739
• Partial differential equations, geometry and stochastic control, in Catalan. Butl. Soc. Catalana Mat. 15 (2000), 7-27
• Existence versus instantaneous blow-up for linear heat equations with singular potentials, with Y. Martel. C. R. Acad. Sci. Paris Ser. I Math. 329 (1999), 973-978
• Stable solutions of semilinear elliptic problems in convex domains, with S. Chanillo. Selecta Math. (New Ser.) 4 (1998), 1-10
• Some simple nonlinear PDE's without solutions, with H. Brezis. Boll. Unione Mat. Ital. 1-B (1998), 223-262
• Weak eigenfunctions for the linearization of extremal elliptic problems, with Y. Martel. J. Funct. Anal. 156 (1998), 30-56
• Nondivergent elliptic equations on manifolds with nonnegative curvature. Comm. Pure Appl. Math. 50 (1997), 623-665
• Truncation of nonlinearities in some supercritical elliptic problems, with P. Majer. C. R. Acad. Sci. Paris Ser. I Math. 322 (1996), 1157-1162
• Regularity for viscosity solutions of fully nonlinear equations F(D^2 u)=0, with L. Caffarelli. Topol. Methods Nonlinear Anal. 6 (1995), 31-48
• On the Alexandroff-Bakelman-Pucci estimate and the reversed Holder inequality for solutions of elliptic and parabolic equations. Comm. Pure Appl. Math. 48 (1995), 539-570

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