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arXiv · 1611.10105

Improvement of flatness for nonlocal phase transitions

Serena Dipierro·Joaquim Serra·Enrico Valdinoci
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Abstract

We prove an improvement of flatness result for nonlocal phase transitions. For a class of nonlocal equations that includes $(-Δ)^{s/2} u = u-u^3$, with~$s\in(0,1)$, we obtain a result in the same spirit of a celebrated theorem of Savin for the equation $-Δu = u-u^3$. As a consequence, we deduce that entire solutions to~$(-Δ)^{s/2} u = u-u^3$ with asymptotically flat level sets are $1$D when~$s\in(0,1)$. 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.

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Serena Dipierro, Joaquim Serra, Enrico Valdinoci. 2017-06-24. Improvement of flatness for nonlocal phase transitions. https://arxiv.org/abs/1611.10105

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