Xavier Fernández-Real

activity

20242026

most citedIntegro-Differential Elliptic Equations

33 citations · 34 across the 15 of their papers we have counts for

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14 papers · 1 filter

math.AP2026

Stable solutions of the Allen--Cahn equation in dimension three are one-dimensional

Hardy Chan, Xavier Fernández-Real, Alessio Figalli +2

We prove that every bounded stable solution of the Allen--Cahn equation in $R^{3}$ is one-dimensional. As a consequence, this shows the validity of De Giorgi's conjecture for…

math.AP2026

The stability conjecture for the Bernoulli problem in dimension four

Xavier Fernández-Real, Joaquim Serra

We prove that every global classical stable solution to the one-phase Bernoulli problem in $R^{4}$ is one-dimensional. In particular, if nonempty, its free boundary consists…

math.AP2026

Superharmonicity of the fractional ground state

Nicola De Nitti, Xavier Fernández-Real, Florian Grube +1

We prove that the positive ground state $u$ of the fractional Laplacian $(- Δ)^{s}$ , $s \in (0, 1)$ , on an arbitrary open set $Ω$ satisfies $- Δ u > λ_{s} (Ω)^{1/ s} u$ in $Ω$ . In one dimension,…

math.AP2026

The fractional Laplacian in Lipschitz domains: Dahlberg's Theorem and $L^{2}$ -solvability

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

Given $s \in (0, 1)$ and a bounded Lipschitz domain $Ω \subset R^{n}$ , we establish a quantitative Dahlberg theory for the $s$ -harmonic measure of $Ω$ , $ω_{s}^{x}$ . In the nonloc…

math.AP2026

Nonlocal minimal surfaces are generically unique, and smooth in one extra dimension

Serena Dipierro, Xavier Fernández-Real, Enrico Valdinoci

We study the fractional Plateau problem for $s$ -minimal sets with prescribed exterior datum. For fixed exterior data, minimizers need not be unique, and singularities may occur bey…

math.AP2026

Classification of blow-ups for the Alt--Phillips problem in three dimensions

Xavier Fernández-Real

We prove a flatness theorem for classical stable homogeneous solutions of the $γ$ -Alt--Phillips free boundary problem in three dimensions in the range $0 < γ \leq 2/3$ , where $γ$ is th…

math.AP2026

Finite index solutions to the Bernoulli problem in three dimensions are axially symmetric

Xavier Fernández-Real, Enric Florit-Simon, Joaquim Serra

We show that every entire solution to the Bernoulli (or one-phase) free boundary problem with finite Morse index in $R^{3}$ is axially symmetric. In fact, we additionally pr…

math.AP2026

Quantitative Convergence of Wasserstein Gradient Flows of Kernel Mean Discrepancies

Lénaïc Chizat, Maria Colombo, Roberto Colombo +1

We study the quantitative convergence of Wasserstein gradient flows of Kernel Mean Discrepancy (KMD) (also known as Maximum Mean Discrepancy (MMD)) functionals. Our setting covers…

math.AP2025

Linearized equation and generic regularity in the Alt-Caffarelli problem

Xavier Fernández-Real, Hui Yu

For the Alt-Caffarelli problem, we study free boundary regularity of energy minimizers. In six dimensions, we show that free boundaries are analytic for generic boundary data. In g…

math.AP2025

Global solutions to the thin obstacle problem with superquadratic growth

Xavier Fernández-Real, Hui Yu

We study rigidity/flexibility properties of global solutions to the thin obstacle problem. For solutions with bounded positive sets, we give a classification in terms of their expa…

math.AP2025

Global Stable Solutions to the Free Boundary Allen--Cahn and Bernoulli Problems in 3D are One-Dimensional

Hardy Chan, Xavier Fernández-Real, Alessio Figalli +1

A long-standing conjecture of De Giorgi asserts that every monotone solution of the Allen--Cahn equation in $R^{n + 1}$ is one-dimensional if $n \leq 7$ . A stronger version…

math.AP2024★ 33 cited

Integro-Differential Elliptic Equations

Xavier Fernández-Real, Xavier Ros-Oton

This book aims to provide a self-contained introduction to the regularity theory for integro-differential elliptic equations, mostly developed in the 21st century. Such a class of…

math.AP2024

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

Xavier Fernández-Real, Florian Gruen

We prove new boundary regularity results for minimizers to the one-phase Alt-Caffarelli functional (also known as Bernoulli free boundary problem) in the case of continuous and Höl…

math.AP2024

Regularity for the Boltzmann equation conditional to pressure and moment bounds

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

We prove that solutions to the Boltzmann equation without cut-off satisfying pointwise bounds on some observables (mass, pressure, and suitable moments) enjoy a uniform bound in $L…

Stable solutions of the Allen--Cahn equation in dimension three are one-dimensional

The stability conjecture for the Bernoulli problem in dimension four

Superharmonicity of the fractional ground state

The fractional Laplacian in Lipschitz domains: Dahlberg's Theorem and L2-solvability

Nonlocal minimal surfaces are generically unique, and smooth in one extra dimension

Classification of blow-ups for the Alt--Phillips problem in three dimensions

Finite index solutions to the Bernoulli problem in three dimensions are axially symmetric

Quantitative Convergence of Wasserstein Gradient Flows of Kernel Mean Discrepancies

Linearized equation and generic regularity in the Alt-Caffarelli problem

Global solutions to the thin obstacle problem with superquadratic growth

Global Stable Solutions to the Free Boundary Allen--Cahn and Bernoulli Problems in 3D are One-Dimensional

Integro-Differential Elliptic Equations

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

Regularity for the Boltzmann equation conditional to pressure and moment bounds

The fractional Laplacian in Lipschitz domains: Dahlberg's Theorem and $L^{2}$ -solvability