33 citations · 34 across the 15 of their papers we have counts for
18 papers
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 is one-dimensional. As a consequence, this shows the validity of De Giorgi's conjecture for…
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 is one-dimensional. In particular, if nonempty, its free boundary consists…
Superharmonicity of the fractional ground state
Nicola De Nitti, Xavier Fernández-Real, Florian Grube +1
We prove that the positive ground state of the fractional Laplacian , , on an arbitrary open set satisfies in . In one dimension,…
The fractional Laplacian in Lipschitz domains: Dahlberg's Theorem and -solvability
Roberto Colombo, Xavier Fernández-Real, Xavier Ros-Oton
Given and a bounded Lipschitz domain , we establish a quantitative Dahlberg theory for the -harmonic measure of , . In the nonloc…
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 -minimal sets with prescribed exterior datum. For fixed exterior data, minimizers need not be unique, and singularities may occur bey…
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 -Alt--Phillips free boundary problem in three dimensions in the range , where is th…
Improvement of flatness in annuli
Xavier Fernández-Real, Enric Florit-Simon, Joaquim Serra
We present a short and flexible improvement-of-flatness argument adapted to the setting of exterior domains, where one is naturally led to work with annuli instead of balls. As a m…
Quantitative Local Convergence of Mean-Field Stein Variational Gradient Flow
Lénaïc Chizat, Maria Colombo, Roberto Colombo +1
Stein Variational Gradient Descent (SVGD) is a deterministic interacting-particle method for sampling from a target probability measure given access to its score function. In the m…
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 is axially symmetric. In fact, we additionally pr…
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…
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…
Convergence of drift-diffusion PDEs arising as Wasserstein gradient flows of convex functions
Lénaïc Chizat, Maria Colombo, Xavier Fernández-Real
We study the quantitative convergence of drift-diffusion PDEs that arise as Wasserstein gradient flows of linearly convex functions over the space of probability measures on ${\mat…
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…
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 is one-dimensional if . A stronger version…
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…
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…
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…
Optimal transport of measures via autonomous vector fields
Nicola De Nitti, Xavier Fernández-Real
We study the problem of transporting one probability measure to another via an autonomous velocity field. We rely on tools from the theory of optimal transport. In one space-dimens…