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Regularity of the nodal set of segregated critical configurations under a weak reflection law

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Abstract

We deal with a class of Lipschitz vector functions U = (u 1, . . . , u h ) whose components are nonnegative, disjointly supported and verify an elliptic equation on each support. Under a weak formulation of a reflection law, related to the Pohoz̆aev identity, we prove that the nodal set is a collection of C 1,α hyper-surfaces (for every 0 < α < 1), up to a residual set with small Hausdorff dimension. This result applies to the asymptotic limits of reaction–diffusion systems with strong competition interactions, to optimal partition problems involving eigenvalues, as well as to segregated standing waves for Bose–Einstein condensates in multiple hyperfine spin states.

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Authors and Affiliations

  1. Faculty of Science, CMAF, University of Lisbon, Av. Prof. Gama Pinto 2, 1649-003, Lisbon, Portugal

    Hugo Tavares

  2. Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano-Bicocca, via Bicocca degli Arcimboldi 8, 20126, Milan, Italy

    Susanna Terracini

Authors
  1. Hugo Tavares
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  2. Susanna Terracini
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Correspondence to Susanna Terracini.

Additional information

Communicated by A. Malchiodi.

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Tavares, H., Terracini, S. Regularity of the nodal set of segregated critical configurations under a weak reflection law. Calc. Var. 45, 273–317 (2012). https://doi.org/10.1007/s00526-011-0458-z

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  • DOI: https://doi.org/10.1007/s00526-011-0458-z

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