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

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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).

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Notes

  1. In order to apply Corollary 4.5 in [41], we observe that assumptions (4.7), (4.8) and (4.10) therein have been already verified. As far as (4.9) is concerned, we recall that as exponent \(\eta >0\) associated to \((-\Delta )^s\) we have to take a positive small number (see [41, Section 2]). Then, for any \(x \in \mathbb {R}^N \setminus B_{r_j}(q_j)\), it results that

    $$\begin{aligned} A \left( 2 \left| 2 \frac{x-q_j}{r_j}\right| ^\eta -1 \right) \geqslant A \left( 2^{1+\eta } -1 \right) \geqslant A \geqslant z^+(x), \end{aligned}$$

    i.e. assumption (4.9) holds.

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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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Authors and Affiliations

  1. School of Mathematics and Statistics, University of Melbourne, 813 Swanston St., Parkville, VIC, 3010, Australia

    Serena Dipierro & Enrico Valdinoci

  2. School of Mathematics and Statistics, University of Western Australia, 35 Stirling Highway, Crawley, Perth, WA, 6009, Australia

    Serena Dipierro & Enrico Valdinoci

  3. Mathematisches Institut, Justus-Liebig-Universität Giessen, Arndtstrasse 2, 35392, Giessen, Germany

    Nicola Soave

  4. Weierstraß-Institut für Angewandte Analysis und Stochastik, Mohrenstraße 39, 10117, Berlin, Germany

    Enrico Valdinoci

  5. Dipartimento di Matematica, Università degli studi di Milano, Via Saldini 50, 20133, Milan, Italy

    Enrico Valdinoci

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  1. Serena Dipierro
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Correspondence to Nicola Soave.

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Dipierro, S., Soave, N. & Valdinoci, E. On fractional elliptic equations in Lipschitz sets and epigraphs: regularity, monotonicity and rigidity results. Math. Ann. 369, 1283–1326 (2017). https://doi.org/10.1007/s00208-016-1487-x

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