Xavier Fernández-Real

I am an Ambizione Fellow at EPFL.

My research focuses in PDEs and Calculus of Variations.

In particular, I have worked in elliptic and parabolic PDE, integro-differential equations, obstacle-type problems, transport equations, etc.

Articles and preprints

• Regularity for the Boltzmann equation conditional to pressure and moment bounds,

  Xavier Fernández-Real, Xavier Ros-Oton, Marvin Weidner

  Submitted, available at arXiv.




• Continuity up to the boundary for minimizers of the one-phase Bernoulli problem,

  Xavier Fernández-Real, Florian Gruen

  Submitted, available at arXiv.




• Optimal transport of measures via autonomous vector fields,

  Nicola De Nitti, Xavier Fernández-Real

  Submitted, available at arXiv.




• Generic properties in free boundary problems,

  Xavier Fernández-Real, Hui Yu

  Submitted, available at arXiv.




• Schauder and Cordes-Nirenberg estimates for nonlocal elliptic equations with singular kernels,

  Xavier Fernández-Real, Xavier Ros-Oton

  Proc. Lond. Math. Soc. 129 (2024), e12629.




• Smooth approximations for fully nonlinear nonlocal elliptic equations,

  Xavier Fernández-Real

  Trans. Amer. Math. Soc. 377 (2024), 495-515.




• Graphical solutions to one-phase free boundary problems,

  Max Engelstein, Xavier Fernández-Real, Hui Yu

  J. Reine Angew. Math. 804 (2023), 155-195.




• Infinite-width limit of deep linear neural networks,

  Lénaïc Chizat, Maria Colombo, Xavier Fernández-Real, Alessio Figalli

  Comm. Pure Appl. Math. 77 (2024), 3958-4007.




• Generic regularity of free boundaries for the thin obstacle problem,

  Xavier Fernández-Real, Clara Torres-Latorre

  Adv. Math. 433 (2023), 109323.




• Optimal regularity for the fully nonlinear thin obstacle problem,

  Maria Colombo, Xavier Fernández-Real, Xavier Ros-Oton

  J. Eur. Math. Soc., to appear.




• Improved regularity of second derivatives for subharmonic functions,

  Xavier Fernández-Real, Riccardo Tione

  Proc. Amer. Math. Soc. 151 (2023), 5283-5297.




• Stable cones in the thin one-phase problem,

  Xavier Fernández-Real, Xavier Ros-Oton

  Amer. J. Math. 146 (2024), 631-685.




• Free boundary regularity for almost every solution to the Signorini problem,

  Xavier Fernández-Real, Xavier Ros-Oton

  Arch. Ration. Mech. Anal. 240 (2021), 419-466.




• On the obstacle problem for the 1D wave equation,

  Xavier Fernández-Real, Alessio Figalli

  Mathematics in Engineering 2 (2020), 584-597.




• On the singular set in the thin obstacle problem: higher order blow-ups and the very thin obstacle problem,

  Xavier Fernández-Real, Yash Jhaveri

  Anal. PDE, 14 (2021), 1599-1669.




• On global solutions to semilinear elliptic equations related to the one-phase free boundary problem,

  Xavier Fernández-Real, Xavier Ros-Oton

  Discrete Contin. Dyn. Syst. A 39 (2019), 6945-6959.




• Regularity of minimal surfaces with lower dimensional obstacles,

  Xavier Fernández-Real, Joaquim Serra

  J. Reine Angew. Math. 767 (2020), 37-75.




• The Lagrangian structure of the Vlasov-Poisson system in domains with specular reflection,

  Xavier Fernández-Real

  Commun. Math. Phys. 364 (2018), 1327-1406.




• The obstacle problem for the fractional Laplacian with critical drift,

  Xavier Fernández-Real, Xavier Ros-Oton

  Math. Ann. 371(3) (2018), 1683-1735.




• C^1,α estimates for the fully nonlinear Signorini problem,

  Xavier Fernández-Real

  Calc. Var. Partial Differential Equations (2016), 55:94.




• Regularity theory for general stable operators: parabolic equations,

  Xavier Fernández-Real, Xavier Ros-Oton

  J. Funct. Anal. 270 (2017), no. 10, 4165-4221.




• Boundary regularity for the fractional heat equation,

  Xavier Fernández-Real, Xavier Ros-Oton

  Rev. Acad. Cienc. Ser. A Math. 110 (2016), 49-64.

Surveys

• The continuous formulation of shallow neural networks as Wasserstein-type gradient flows,

  Xavier Fernández-Real, Alessio Figalli

  In: Analysis at large, Springer 2022.




• The thin obstacle problem: a survey,

  Xavier Fernández-Real

  Publ. Mat. 66 (2022), 3-55.

Books

• 

  
  
  Integro-Differential Elliptic Equations

  Xavier Fernández-Real, Xavier Ros-Oton

  Progress in Mathematics, vol. 350, Birkhäuser, 2024.

  Last updated: 2024/11/21.



• 

  
  Regularity Theory for Elliptic PDE

  Xavier Fernández-Real, Xavier Ros-Oton

  Zurich Lectures in Advanced Mathematics , EMS Press, 2022.

  Last updated: 2023/10/10.

  
  AMS Book Review

Theses

• PhD Thesis (see ETH Research Collection)


• Bachelor's Degree Thesis (see UPCommons)