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122 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…
Sharp stability of Alexandrov's theorem for domains in the small-excess regime
Alessio Figalli, Yi Ru-Ya Zhang
We prove a sharp quantitative stability result for Alexandrov's theorem in arbitrary dimension for bounded open sets in a small-excess regime. More precisely, if $E\subset \m…
Stable Semilinear Elliptic Equations: -Regularity à la Brezis and Dimensional Bounds for the Singular Set
Alessio Figalli, Federico Franceschini
We develop a quantitative partial regularity theory for stable solutions of \[ -Δu=f(u), \] where is increasing and convex. The theory is uniform in t…
A dimension-free interpolation of Caffarelli's contraction theorem
Bader Ammari, Alessio Figalli
We prove global Lipschitz estimates for Brenier maps between probability measures on whose densities belong to the family $$ ρ_{U,\,p}=Z_{U,\, p}^{-1}\exp(-Θ_p(U)),…
Global regularity and free boundary geometry in the planar Choné-Rochet model
Shibing Chen, Alessio Figalli, Yi Ru-Ya Zhang
In this paper, we study minimizers of the Choné--Rochet variational problem in dimension two. We first establish global regularity on arbitrary bounded convex domains, and th…
An anisotropic Serrin's problem in general domains
Alessio Figalli, Yi Ru-Ya Zhang
Serrin's symmetry theorem shows that the classical overdetermined torsion problem forces the domain to be a ball. Extending this rigidity statement to merely Lipschitz (and more ge…