Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter January 27, 2023

Characterizations of the viscosity solution of a nonlocal and nonlinear equation induced by the fractional p-Laplace and the fractional p-convexity

  • Shaoguang Shi EMAIL logo , Zhichun Zhai and Lei Zhang EMAIL logo

Abstract

In this paper, when studying the connection between the fractional convexity and the fractional p-Laplace operator, we deduce a nonlocal and nonlinear equation. Firstly, we will prove the existence and uniqueness of the viscosity solution of this equation. Then we will show that u ( x ) is the viscosity sub-solution of the equation if and only if u ( x ) is so-called ( α , p ) -convex. Finally, we will characterize the viscosity solution of this equation as the envelope of an ( α , p ) -convex sub-solution. The technique involves attainability of the exterior datum and a comparison principle for the nonlocal and nonlinear equation.


Communicated by Guofang Wang


Award Identifier / Grant number: 12271232

Award Identifier / Grant number: 12071197

Award Identifier / Grant number: 2020KJI002

Award Identifier / Grant number: ZR2021MA079

Funding statement: Shaoguang Shi and Lei Zhang were supported by Natural Science Foundation of China (No. 12271232, No. 12071197) and Natural Science Foundation of Shandong Province (No. 2020KJI002, No. ZR2021MA079).

Acknowledgements

The authors wish to thank the anonymous referees cordially for his/her comments and for providing additional relevant references.

References

[1] S. Angenent, P. Daskalopoulos and N. Sesum, Uniqueness of two-convex closed ancient solutions to the mean curvature flow, Ann. of Math. (2) 192 (2020), no. 2, 353–436. 10.4007/annals.2020.192.2.2Search in Google Scholar

[2] J. M. Ball and G.-Q. G. Chen, Entropy and convexity for nonlinear partial differential equations, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 371 (2013), no. 2005, Article ID 20120340. 10.1098/rsta.2012.0340Search in Google Scholar PubMed PubMed Central

[3] G. Barles, E. Chasseigne and C. Imbert, On the Dirichlet problem for second-order elliptic integro-differential equations, Indiana Univ. Math. J. 57 (2008), no. 1, 213–246. 10.1512/iumj.2008.57.3315Search in Google Scholar

[4] G. Barles and C. Imbert, Second-order elliptic integro-differential equations: viscosity solutions’ theory revisited, Ann. Inst. H. Poincaré C Anal. Non Linéaire 25 (2008), no. 3, 567–585. 10.1016/j.anihpc.2007.02.007Search in Google Scholar

[5] I. Birindelli, G. Galise and D. Schiera, Maximum principles and related problems for a class of nonlocal extremal operators, Ann. Mat. Pura Appl. (4) 201 (2022), no. 5, 2371–2412. 10.1007/s10231-022-01203-zSearch in Google Scholar

[6] I. Birindelli, G. Galise and E. Topp, Fractional truncated Laplacians: Representation formula, fundamental solutions and applications, NoDEA Nonlinear Differential Equations Appl. 29 (2022), no. 3, Paper No. 26. 10.1007/s00030-022-00757-4Search in Google Scholar

[7] P. Blanc and J. D. Rossi, Games for eigenvalues of the Hessian and concave/convex envelopes, J. Math. Pures Appl. (9) 127 (2019), 192–215. 10.1016/j.matpur.2018.08.007Search in Google Scholar

[8] L. Brasco and E. Lindgren, Higher Sobolev regularity for the fractional p-Laplace equation in the superquadratic case, Adv. Math. 304 (2017), 300–354. 10.1016/j.aim.2016.03.039Search in Google Scholar

[9] M. L. de Borbón, L. M. Del Pezzo and P. Ochoa, Weak and viscosity solutions for non-homogeneous fractional equations in Orlicz spaces, Adv. Differential Equations 27 (2022), no. 11–12, 735–780. 10.57262/ade027-1112-735Search in Google Scholar

[10] L. M. Del Pezzo and A. Quaas, A Hopf’s lemma and a strong minimum principle for the fractional p-Laplacian, J. Differential Equations 263 (2017), no. 1, 765–778. 10.1016/j.jde.2017.02.051Search in Google Scholar

[11] L. M. Del Pezzo and A. Quaas, Non-resonant Fredholm alternative and anti-maximum principle for the fractional p-Laplacian, J. Fixed Point Theory Appl. 19 (2017), no. 1, 939–958. 10.1007/s11784-017-0405-5Search in Google Scholar

[12] L. M. Del Pezzo, A. Quaas and J. D. Rossi, Fractional convexity, Math. Ann. 383 (2022), no. 3–4, 1687–1719. 10.1007/s00208-021-02254-ySearch in Google Scholar

[13] L. M. Del Pezzo and J. D. Rossi, Eigenvalues for systems of fractional p-Laplacians, Rocky Mountain J. Math. 48 (2018), no. 4, 1077–1104. 10.1216/RMJ-2018-48-4-1077Search in Google Scholar

[14] A. Di Castro, T. Kuusi and G. Palatucci, Local behavior of fractional p-minimizers, Ann. Inst. H. Poincaré C Anal. Non Linéaire 33 (2016), no. 5, 1279–1299. 10.1016/j.anihpc.2015.04.003Search in Google Scholar

[15] R. J. Dwilewicz, A short history of convexity, Differ. Geom. Dyn. Syst. 11 (2009), 112–129. Search in Google Scholar

[16] P. Guan and X.-N. Ma, The Christoffel–Minkowski problem. I. Convexity of solutions of a Hessian equation, Invent. Math. 151 (2003), no. 3, 553–577. 10.1007/s00222-002-0259-2Search in Google Scholar

[17] F. R. Harvey and H. B. Lawson, Jr., Dirichlet duality and the nonlinear Dirichlet problem, Comm. Pure Appl. Math. 62 (2009), no. 3, 396–443. 10.1002/cpa.20265Search in Google Scholar

[18] J. Korvenpää, T. Kuusi and E. Lindgren, Equivalence of solutions to fractional p-Laplace type equations, J. Math. Pures Appl. (9) 132 (2019), 1–26. 10.1016/j.matpur.2017.10.004Search in Google Scholar

[19] E. Lindgren and P. Lindqvist, Fractional eigenvalues, Calc. Var. Partial Differential Equations 49 (2014), no. 1–2, 795–826. 10.1007/s00526-013-0600-1Search in Google Scholar

[20] A. M. Oberman, The convex envelope is the solution of a nonlinear obstacle problem, Proc. Amer. Math. Soc. 135 (2007), no. 6, 1689–1694. 10.1090/S0002-9939-07-08887-9Search in Google Scholar

[21] A. M. Oberman and L. Silvestre, The Dirichlet problem for the convex envelope, Trans. Amer. Math. Soc. 363 (2011), no. 11, 5871–5886. 10.1090/S0002-9947-2011-05240-2Search in Google Scholar

[22] R. T. Rockafellar, Convex Analysis, Princeton Math. Ser. 28, Princeton University, Princeton, 1970. Search in Google Scholar

[23] S. Shi, Some notes on supersolutions of fractional p-Laplace equation, J. Math. Anal. Appl. 463 (2018), no. 2, 1052–1074. 10.1016/j.jmaa.2018.03.064Search in Google Scholar

[24] S. Shi and J. Xiao, A tracing of the fractional temperature field, Sci. China Math. 60 (2017), no. 11, 2303–2320. 10.1007/s11425-016-0494-6Search in Google Scholar

[25] S. Shi and J. Xiao, Fractional capacities relative to bounded open Lipschitz sets complemented, Calc. Var. Partial Differential Equations 56 (2017), no. 1, Paper No. 3. 10.1007/s00526-016-1105-5Search in Google Scholar

[26] S. Shi, L. Zhang and G. Wang, Fractional non-linear regularity, potential and balayage, J. Geom. Anal. 32 (2022), no. 8, Paper No. 221. 10.1007/s12220-022-00956-6Search in Google Scholar

[27] M. L. J. van de Vel, Theory of Convex Structures, North-Holland Math. Libr. 50, North-Holland, Amsterdam, 1993. Search in Google Scholar

[28] A. Weinstein, Periodic orbits for convex Hamiltonian systems, Ann. of Math. (2) 108 (1978), no. 3, 507–518. 10.2307/1971185Search in Google Scholar

Received: 2021-12-26
Revised: 2022-08-08
Accepted: 2022-12-15
Published Online: 2023-01-27
Published in Print: 2024-01-01

© 2023 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 20.5.2024 from https://www.degruyter.com/document/doi/10.1515/acv-2021-0110/html
Scroll to top button