Abstract
For the singularly perturbed system
we prove that flat segregated interfaces are uniformly Lipschitz as \(\beta \rightarrow +\infty \). As a byproduct of the proof we also obtain the optimal lower bound near flat interfaces,
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References
Berestycki, H., Lin, T., Wei, J., Zhao, C.: On phase-separation model: asymptotics and qualitative properties. Arch. Ration. Mech. Anal. 208(1), 163–200 (2013)
Berestycki, H., Terracini, S., Wang, K., Wei, J.: Existence and stability of entire solutions of an elliptic system modeling phase separation. Adv. Math. 243, 102–126 (2013)
Caffarelli, L., Cordoba, A.: Phase transitions: uniform regularity of the intermediate layers. J. fur die reine und Angew. Math. (Crelles J.) 593, 209–235 (2006)
Caffarelli, L., Lin, F.: Singularly perturbed elliptic systems and multi-valued harmonic functions with free boundaries. J. Am. Math. Soc. 21, 847–862 (2008)
Dancer, E.N., Wang, K., Zhang, Z.: The limit equation for the Gross–Pitaevskii equations and S. Terracini’s conjecture. J. Funct. Anal. 262(2), 1087–1131 (2012)
Soave, N., Zilio, A.: Uniform bounds for strongly competing systems: the optimal Lipschitz case. Arch. Ration. Mech. Anal. 218(2), 647–697 (2015)
Soave, N., Zilio, A.: On phase separation in systems of coupled elliptic equations: asymptotic analysis and geometric aspects. Ann. de l’Institut Henri Poincare (C) Non Linear Anal. 34(3), 625–654 (2017)
Tavares, H., Terracini, S.: Regularity of the nodal set of segregated critical configurations under a weak reflection law. Calc. Var. PDEs 45(3–4), 273–317 (2012)
Wang, K.: On the De Giorgi type conjecture for an elliptic system modeling phase separation. Commun. PDE 39(4), 696–739 (2014)
Wang, K.: Harmonic apparoximation and improvement of flatness in a singularly perturbation problem. Manuscr. Math. 146(1–2), 281–298 (2015)
Wang, K.: A new proof of Savin’s theorem on Allen–Cahn equations. J. Eur. Math. Soc (to appear)
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Communicated by F. H. Lin.
The author’s research was partially supported by “the Fundamental Research Funds for the Central Universities” and NSFC No. 11301522. I would like to thank the refree for valuable suggestions.
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Wang, K. Uniform Lipschitz regularity of flat segregated interfaces in a singularly perturbed problem. Calc. Var. 56, 135 (2017). https://doi.org/10.1007/s00526-017-1235-4
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DOI: https://doi.org/10.1007/s00526-017-1235-4