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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…
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…
Phase transitions with bounded index: Parallels to De Giorgi's conjecture
Enric Florit-Simon
A well-known conjecture of De Giorgi -- motivated by analogy with the Bernstein problem for minimal surfaces -- asserts the rigidity of monotone solutions to the Allen--Cahn equati…
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…
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…