Xavier Fernández-Real

activity

20142026

most citedRegularity Theory for Elliptic PDE

76 citations · 113 across the 14 of their papers we have counts for

collaborators

21 papers

math.AP2026

Bellman Equations with Sub-Lipschitz Hessians

Xavier Fernández-Real

We construct homogeneous solutions with non-Lipschitz Hessian for finite, constant-coefficient Bellman equations. First, for every $σ \in (0, 1)$ , we find two uniformly elliptic matri…

math.AP2023

Generic properties in free boundary problems

Xavier Fernández-Real, Hui Yu

In this work, we show the generic uniqueness of minimizers for a large class of energies, including the Alt-Caffarelli and Alt-Phillips functionals. We then prove the generic regul…

math.AP2023

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

Xavier Fernández-Real, Xavier Ros-Oton

We study integro-differential elliptic equations (of order $2 s$ ) with variable coefficients, and prove the natural and most general Schauder-type estimates that can hold in this se…

math.AP2023★ 2 cited

Smooth approximations for fully nonlinear nonlocal elliptic equations

Xavier Fernández-Real

We show that any viscosity solution to a general fully nonlinear nonlocal elliptic equation can be approximated by smooth ( $C^{\infty}$ ) solutions.

math.AP2023★ 76 cited

Regularity Theory for Elliptic PDE

Xavier Fernández-Real, Xavier Ros-Oton

This manuscript aims to provide a self-contained introduction to the regularity theory for elliptic PDE, focusing on the main ideas rather than proving all results in their greates…

math.AP2022★ 1 cited

Graphical solutions to one-phase free boundary problems

Max Engelstein, Xavier Fernández-Real, Hui Yu

We study viscosity solutions to the classical one-phase problem and its thin counterpart. In low dimensions, we show that when the free boundary is the graph of a continuous functi…

cs.LG2022★ 5 cited

Infinite-width limit of deep linear neural networks

Lénaïc Chizat, Maria Colombo, Xavier Fernández-Real +1

This paper studies the infinite-width limit of deep linear neural networks initialized with random parameters. We obtain that, when the number of neurons diverges, the training dyn…

math.AP2022

Generic regularity of free boundaries for the thin obstacle problem

Xavier Fernández-Real, Clara Torres-Latorre

The free boundary for the Signorini problem in $R^{n + 1}$ is smooth outside of a degenerate set, which can have the same dimension ( $n - 1$ ) as the free boundary itself. In […

math.AP2021

Optimal regularity for the fully nonlinear thin obstacle problem

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

In this work we establish the optimal regularity for solutions to the fully nonlinear thin obstacle problem. In particular, we show the existence of an optimal exponent $α_{F}$ such…

math.AP2021

Improved regularity of second derivatives for subharmonic functions

Xavier Fernández-Real, Riccardo Tione

In this note, we prove that if a subharmonic function $Δ u \geq 0$ has pure second derivatives $\partial_{ii} u$ that are signed measures, then their negative part $(\partial_{ii} u)_…

math.AP2020

The Thin Obstacle Problem: A Survey

Xavier Fernández-Real

In this work we present a general introduction to the Signorini problem (or thin obstacle problem). It is a self-contained survey that aims to cover the main currently known result…

math.AP2020★ 1 cited

Stable cones in the thin one-phase problem

Xavier Fernández-Real, Xavier Ros-Oton

The aim of this work is to study homogeneous stable solutions to the thin (or fractional) one-phase free boundary problem. The problem of classifying stable (or minimal) homogeneou…

math.AP2019

On the obstacle problem for the 1D wave equation

Xavier Fernández-Real, Alessio Figalli

Our goal is to review the known theory on the one-dimensional obstacle problem for the wave equation, and to discuss some extensions. We introduce the setting established by Schatz…

math.AP2019★ 12 cited

Free boundary regularity for almost every solution to the Signorini problem

Xavier Fernández-Real, Xavier Ros-Oton

We investigate the regularity of the free boundary for the Signorini problem in $R^{n + 1}$ . It is known that regular points are $(n - 1)$ -dimensional and $C^{\infty}$ . However,…

math.AP2018★ 10 cited

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

Xavier Fernández-Real, Yash Jhaveri

In this work, we consider the singular set in the thin obstacle problem with weight $∣ x_{n + 1} ∣^{a}$ for $a \in (- 1, 1)$ , which arises as the local extension of the obstacle problem fo…

math.AP2018

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

Xavier Fernández-Real, Xavier Ros-Oton

Motivated by its relation to models of flame propagation, we study globally Lipschitz solutions of $Δ u = f (u)$ in $R^{n}$ , where $f$ is smooth, non-negative, with support in t…

math.AP2018

Regularity of minimal surfaces with lower dimensional obstacles

Xavier Fernández-Real, Joaquim Serra

We study the Plateau problem with a lower dimensional obstacle in $R^{n}$ . Intuitively, in $R^{3}$ this corresponds to a soap film (spanning a given contour) that is…

math.AP2017★ 6 cited

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

Xavier Fernández-Real

In this work, we deal with the Vlasov-Poisson system in smooth physical domains with specular boundary condition, under mild integrability assumptions, and $d \geq 3$ . We show that…

math.AP2016

The obstacle problem for the fractional Laplacian with critical drift

Xavier Fernández-Real, Xavier Ros-Oton

We study the obstacle problem for the fractional Laplacian with drift, $min {(- Δ)^{s} u + b \cdot \nabla u, u - φ} = 0$ in $R^{n}$ , in the critical regime $s = \f…

math.AP2015

Regularity theory for general stable operators: parabolic equations

Xavier Fernández-Real, Xavier Ros-Oton

We establish sharp interior and boundary regularity estimates for solutions to $\partial_{t} u - Lu = f (t, x)$ in $I \times Ω$ , with $I \subset R$ and $Ω \subset R^{n}$ …

math.AP2014

Boundary regularity for the fractional heat equation

Xavier Fernández-Real, Xavier Ros-Oton

We study the regularity up to the boundary of solutions to fractional heat equation in bounded $C^{1, 1}$ domains. More precisely, we consider solutions to $\partial_t u + (-Δ)^s u=…