6 papers · 1 filter
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…
-convergence for higher order nonlocal phase transitions
Hardy Chan, Serena Dipierro, Mattia Freguglia +2
For every , we study the asymptotic behavior of the -rescaled sum of the -fractional Allen-Cahn energy and the squared -norm of its first variation…
Convergence of the fractional Yamabe flow for arbitrary initial energy
Jingeon An-Lacroix, Hardy Chan, Pak Tung Ho
Since the seminal paper of Graham and Zworski (Invent. Math. 2003), conformal geometric problems are studied in the fractional setting. We consider the convergence of fractional Ya…
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…
Spectral properties of Lévy Fokker--Planck equations
Hardy Chan, Marco A. Fontelos, María del Mar González
Hermite polynomials, which are associated to a Gaussian weight and solve the Laplace equation with a drift term of linear growth, are classical in analysis and well-understood via…
-Limsup estimate for a nonlocal approximation of the Willmore functional
Hardy Chan, Mattia Freguglia, Marco Inversi
We propose a possible nonlocal approximation of the Willmore functional, in the sense of Gamma-convergence, based on the first variation ot the fractional Allen-Cahn energies, and…