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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…
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…
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…
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…