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Research Article

A fractional Hopf Lemma for sign-changing solutions

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Pages 217-241 | Received 11 Jun 2023, Accepted 20 Mar 2024, Published online: 25 Apr 2024

References

  • Gilbarg, D., Trudinger, N. S. (1983). Elliptic Partial Differential Equations of Second Order, volume 224 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 2nd ed. Berlin: Springer-Verlag.
  • Greco, A., Servadei, R. (2016). Hopf’s lemma and constrained radial symmetry for the fractional Laplacian. Math. Res. Lett. 23(3):863–885. DOI: 10.4310/MRL.2016.v23.n3.a14.
  • Ros-Oton, X. (2016). Nonlocal elliptic equations in bounded domains: a survey. Publ. Mat. 60(1):3–26. DOI: 10.5565/PUBLMAT_60116_01.
  • Fall, M. M., Jarohs, S. (2015). Overdetermined problems with fractional Laplacian. ESAIM Control Optim. Calc. Var. 21(4):924–938. DOI: 10.1051/cocv/2014048.
  • Soave, N., Valdinoci, E. (2019). Overdetermined problems for the fractional Laplacian in exterior and annular sets. J. Anal. Math. 137(1):101–134. DOI: 10.1007/s11854-018-0067-2.
  • Ciraolo, G., Dipierro, S., Poggesi, G., Pollastro, L., Valdinoci, E. (2023). Symmetry and quantitative stability for the parallel surface fractional torsion problem. Trans. Amer. Math. Soc. 376(5):3515–3540. DOI: 10.1090/tran/8837.
  • Dipierro, S., Savin, O., Valdinoci, E. (2017). All functions are locally s-harmonic up to a small error. J. Eur. Math. Soc. (JEMS) 19(4):957–966. DOI: 10.4171/jems/684.
  • Kaßmann, M. (2000). Harnack-Ungleichungen für nichtlokale Differentialoperatoren und Dirichlet-Formen, volume 336 of Bonner Mathematische Schriften [Bonn Mathematical Publications]. Universität Bonn, Mathematisches Institut, Bonn, 2001. Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn.
  • Kaßmann, M. (2011). A new formulation of Harnack’s inequality for nonlocal operators. C. R. Math. Acad. Sci. Paris 349(11–12):637–640. DOI: 10.1016/j.crma.2011.04.014.
  • Bucur, C., Valdinoci, E. (2016). Nonlocal Diffusion and Applications, volume 20 of Lecture Notes of the Unione Matematica Italiana. Cham: Springer; Unione Matematica Italiana, Bologna.
  • Cora, G., Iacopetti, A. (2018). On the structure of the nodal set and asymptotics of least energy sign-changing radial solutions of the fractional Brezis-Nirenberg problem. Nonlinear Anal. 176:226–271. DOI: 10.1016/j.na.2018.07.001.
  • Ros-Oton, X., Serra, J. (2016). Regularity theory for general stable operators. J. Differ. Equ. 260(12):8675–8715. DOI: 10.1016/j.jde.2016.02.033.
  • Ros-Oton, X., Serra, J., Valdinoci, E.(2017). Pohozaev identities for anisotropic integrodifferential operators. Commun. Partial Differ. Equ. 42(8):1290–1321. DOI: 10.1080/03605302.2017.1349148.
  • Dalibard, A.-L., Gérard-Varet, D. (2013). On shape optimization problems involving the fractional Laplacian. ESAIM Control Optim. Calc. Var. 19(4):976–1013. DOI: 10.1051/cocv/2012041.
  • Dipierro, S., Soave, N., Valdinoci, E. (2017). On fractional elliptic equations in Lipschitz sets and epigraphs: regularity, monotonicity and rigidity results. Math. Ann. 369(3–4):1283–1326. DOI: 10.1007/s00208-016-1487-x.
  • Alvarado, R., Brigham, D., Maz’ya, V. G., Mitrea, M., Ziadé, E. (2011). On the regularity of domains satisfying a uniform hour-glass condition and a sharp version of the Hopf-Oleinik boundary point principle. J. Math. Sci. 176(3):281–360. DOI: 10.1007/s10958-011-0398-3.
  • Ros-Oton, X., Serra, J. (2017). Boundary regularity estimates for nonlocal elliptic equations in C1 and C1,n domains. Ann. Mat. Pura Appl. (4) 196(5):1637–1668. DOI: 10.1007/s10231-016-0632-1.
  • De Luca, A., Felli, V., Vita, S. (2022). Strong unique continuation and local asymptotics at the boundary for fractional elliptic equations. Adv. Math. 400:Paper No. 108279, 67. DOI: 10.1016/j.aim.2022.108279.
  • Ros-Oton, X., Serra, J. (2014). The Pohozaev identity for the fractional Laplacian. Arch. Ration. Mech. Anal. 213(2):587–628. DOI: 10.1007/s00205-014-0740-2.
  • Caffarelli, L., Silvestre, L. (2007). An extension problem related to the fractional Laplacian. Commun. Partial Differ. Equ. 32(7–9):1245–1260. DOI: 10.1080/03605300600987306.
  • Fall, M. M., Felli, V. (2014). Unique continuation property and local asymptotics of solutions to fractional elliptic equations. Commun. Partial Differ. Equ. 39(2):354–397. DOI: 10.1080/03605302.2013.825918.
  • Rüland, A. (2015). Unique continuation for fractional Schrödinger equations with rough potentials. Commun. Partial Differ. Equ. 40(1):77–114. DOI: 10.1080/03605302.2014.905594.
  • Dipierro, S., Savin, O., Valdinoci, E. (2019). Local approximation of arbitrary functions by solutions of nonlocal equations. J. Geom. Anal. 29(2):1428–1455. DOI: 10.1007/s12220-018-0045-z.
  • Bucur,C. (2016). Some observations on the Green function for the ball in the fractional Laplace framework. Commun. Pure Appl. Anal. 15(2):657–699. DOI: 10.3934/cpaa.2016.15.657.

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