5 papers · 1 filter
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…
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…
On stable solutions to the Allen-Cahn equation with bounded energy density in
Enric Florit-Simon, Joaquim Serra
We show that stable solutions to the Allen-Cahn equation with bounded energy density (or equivalently, with cubic energy growth) are one-dimensional. Thi…
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 \(\mathbb{R}^{n+1}\) is one-dimensional if \(n \leq 7\). A stronger vers…
Fractional Sobolev spaces on Riemannian manifolds
Michele Caselli, Enric Florit-Simon, Joaquim Serra
This article studies the canonical Hilbert energy on a Riemannian manifold for , with particular focus on the case of closed manifolds. Several equivalent d…