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Mathematische Annalen

, Volume 369, Issue 3–4, pp 1283–1326 | Cite as

On fractional elliptic equations in Lipschitz sets and epigraphs: regularity, monotonicity and rigidity results

  • Serena Dipierro
  • Nicola SoaveEmail author
  • Enrico Valdinoci
Article

Abstract

We consider a nonlocal equation set in an unbounded domain with the epigraph property. We prove symmetry, monotonicity and rigidity results. In particular, we deal with halfspaces, coercive epigraphs and epigraphs that are flat at infinity. These results can be seen as the nonlocal counterpart of the celebrated article (Berestycki et al., Commun Pure Appl Math 50(11):1089–1111, 1997).

Notes

Acknowledgments

In a preliminary version of this paper (see [22]), the proof of Lemma 3.2 was unnecessarily complicated: we are indebted to Mouhamed Moustapha Fall for the simpler argument that we incorporated in the present version of this paper. Part of this work was carried out while Serena Dipierro and Enrico Valdinoci were visiting the Justus-Liebig-Universität Giessen, which they wish to thank for the hospitality. This work has been supported by the Alexander von Humboldt Foundation, the ERC grant 277749 E.P.S.I.L.O.N. “Elliptic Pde’s and Symmetry of Interfaces and Layers for Odd Nonlinearities”, the PRIN grant 201274FYK7 “Aspetti variazionali e perturbativi nei problemi differenziali nonlineari” and the ERC grant 339958 Com.Pat. “Complex Patterns for Strongly Interacting Dynamical Systems”.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsUniversity of MelbourneParkvilleAustralia
  2. 2.School of Mathematics and StatisticsUniversity of Western AustraliaPerthAustralia
  3. 3.Mathematisches InstitutJustus-Liebig-Universität GiessenGiessenGermany
  4. 4.Weierstraß-Institut für Angewandte Analysis und StochastikBerlinGermany
  5. 5.Dipartimento di MatematicaUniversità degli studi di MilanoMilanItaly

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