Improvement of flatness for nonlocal phase transitions

2016/11/30 by Dipierro, Serena, Serra, Joaquim, Valdinoci, Enrico · 2 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1611.10105

Abstract

We prove an improvement of flatness result for nonlocal phase transitions. For a class of nonlocal equations that includes (-Δ)^s/2 u = u-u³, with~s∈(0,1), we obtain a result in the same spirit of a celebrated theorem of Savin for the equation -Δu = u-u³. As a consequence, we deduce that entire solutions to~(-Δ)^s/2 u = u-u³ with asymptotically flat level sets are 1D when~s∈(0,1). The results presented are completely new even for the case of the fractional Laplacian, but the robustness of the proofs allows us to treat also more general, possibly anisotropic, integro-differential operators.

Cited by

• Phase transitions in two-component Bose-Einstein condensates with Rabi frequency (II): The De Giorgi conjecture for the nonlocal problem in ℝ² or ℝ³
• Phase transitions in two-component Bose-Einstein condensates with Rabi frequency (I): The De Giorgi conjecture for the local problem in ℝ³

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