On the regularity theory for mixed local and nonlocal quasilinear elliptic equations

Garain, Prashanta; Kinnunen, Juha

doi:10.1090/tran/8621

On the regularity theory for mixed local and nonlocal quasilinear elliptic equations
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Trans. Amer. Math. Soc. 375 (2022), 5393-5423 Request permission

Abstract:

We consider a combination of local and nonlocal $p$-Laplace equations and discuss several regularity properties of weak solutions. More precisely, we establish local boundedness of weak subsolutions, local Hölder continuity of weak solutions, Harnack inequality for weak solutions and weak Harnack inequality for weak supersolutions. We also discuss lower semicontinuity of weak supersolutions as well as upper semicontinuity of weak subsolutions. Our approach is purely analytic and it is based on the De Giorgi-Nash-Moser theory, the expansion of positivity and estimates involving a tail term. The main results apply to sign changing solutions and capture both local and nonlocal features of the equation.

References

Siva Athreya and Koushik Ramachandran, Harnack inequality for non-local Schrödinger operators, Potential Anal. 48 (2018), no. 4, 515–551. MR 3794384, DOI 10.1007/s11118-017-9646-6
Agnid Banerjee, Prashanta Garain, and Juha Kinnunen. Lower semicontinuity and pointwise behavior of supersolutions for some doubly nonlinear nonlocal parabolic $p$-Laplace equations, arXiv:2101.10042, January 2021.
Martin T. Barlow, Richard F. Bass, Zhen-Qing Chen, and Moritz Kassmann, Non-local Dirichlet forms and symmetric jump processes, Trans. Amer. Math. Soc. 361 (2009), no. 4, 1963–1999. MR 2465826, DOI 10.1090/S0002-9947-08-04544-3
Stefano Biagi, Serena Dipierro, Enrico Valdinoci, and Eugenio Vecchi. Mixed local and nonlocal elliptic operators: regularity and maximum principles. arXiv e-prints, page arXiv:2005.06907, May 2020.
Stefano Biagi, Eugenio Vecchi, Serena Dipierro, and Enrico Valdinoci, Semilinear elliptic equations involving mixed local and nonlocal operators, Proc. Roy. Soc. Edinburgh Sect. A 151 (2021), no. 5, 1611–1641. MR 4313576, DOI 10.1017/prm.2020.75
Stefano Biagi, Serena Dipierro, Enrico Valdinoci, and Eugenio Vecchi. A Hong-Krahn-Szegö inequality for mixed local and nonlocal operators. arXiv e-prints, page arXiv:2110.07129, October 2021.
Verena Bögelein, Frank Duzaar, and Naian Liao, On the Hölder regularity of signed solutions to a doubly nonlinear equation, J. Funct. Anal. 281 (2021), no. 9, Paper No. 109173, 58. MR 4287785, DOI 10.1016/j.jfa.2021.109173
Lorenzo Brasco and Erik Lindgren, Higher Sobolev regularity for the fractional $p$-Laplace equation in the superquadratic case, Adv. Math. 304 (2017), 300–354. MR 3558212, DOI 10.1016/j.aim.2016.03.039
S. Buccheri, J. V. da Silva, and L. H. de Miranda. A system of local/nonlocal $p$-Laplacians: the eigenvalue problem and its asymptotic limit as $p\to \infty$. arXiv:2001.05985, January 2020.
Zhen-Qing Chen, Panki Kim, and Renming Song, Heat kernel estimates for $\Delta +\Delta ^{\alpha /2}$ in $C^{1,1}$ open sets, J. Lond. Math. Soc. (2) 84 (2011), no. 1, 58–80. MR 2819690, DOI 10.1112/jlms/jdq102
Zhen-Qing Chen, Panki Kim, Renming Song, and Zoran Vondraček, Sharp Green function estimates for $\Delta +\Delta ^{\alpha /2}$ in $C^{1,1}$ open sets and their applications, Illinois J. Math. 54 (2010), no. 3, 981–1024 (2012). MR 2928344
Zhen-Qing Chen, Panki Kim, Renming Song, and Zoran Vondraček, Boundary Harnack principle for $\Delta +\Delta ^{\alpha /2}$, Trans. Amer. Math. Soc. 364 (2012), no. 8, 4169–4205. MR 2912450, DOI 10.1090/S0002-9947-2012-05542-5
Zhen-Qing Chen and Takashi Kumagai, A priori Hölder estimate, parabolic Harnack principle and heat kernel estimates for diffusions with jumps, Rev. Mat. Iberoam. 26 (2010), no. 2, 551–589. MR 2677007, DOI 10.4171/RMI/609
Agnese Di Castro, Tuomo Kuusi, and Giampiero Palatucci, Nonlocal Harnack inequalities, J. Funct. Anal. 267 (2014), no. 6, 1807–1836. MR 3237774, DOI 10.1016/j.jfa.2014.05.023
Agnese Di Castro, Tuomo Kuusi, and Giampiero Palatucci, Local behavior of fractional $p$-minimizers, Ann. Inst. H. Poincaré C Anal. Non Linéaire 33 (2016), no. 5, 1279–1299. MR 3542614, DOI 10.1016/j.anihpc.2015.04.003
Eleonora Di Nezza, Giampiero Palatucci, and Enrico Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), no. 5, 521–573. MR 2944369, DOI 10.1016/j.bulsci.2011.12.004
Emmanuele DiBenedetto, Degenerate parabolic equations, Universitext, Springer-Verlag, New York, 1993. MR 1230384, DOI 10.1007/978-1-4612-0895-2
Serena Dipierro, Edoardo Proietti Lippi, and Enrico Valdinoci, Linear theory for a mixed operator with Neumann conditions, arXiv:2006.03850, June 2020.
Serena Dipierro, Edoardo Proietti Lippi, and Enrico Valdinoci, (Non)local logistic equations with Neumann conditions, arXiv:2101.02315, January 2021.
Serena Dipierro, Xavier Ros-Oton, Joaquim Serra, and Enrico Valdinoci, Non-symmetric stable operators: regularity theory and integration by parts, arXiv:2012.04833, December 2020.
Serena Dipierro, Ovidiu Savin, and Enrico Valdinoci, All functions are locally $s$-harmonic up to a small error, J. Eur. Math. Soc. (JEMS) 19 (2017), no. 4, 957–966. MR 3626547, DOI 10.4171/JEMS/684
Lawrence C. Evans, Partial differential equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. MR 1625845
Matthieu Felsinger and Moritz Kassmann, Local regularity for parabolic nonlocal operators, Comm. Partial Differential Equations 38 (2013), no. 9, 1539–1573. MR 3169755, DOI 10.1080/03605302.2013.808211
Mohammud Foondun, Heat kernel estimates and Harnack inequalities for some Dirichlet forms with non-local part, Electron. J. Probab. 14 (2009), no. 11, 314–340. MR 2480543, DOI 10.1214/EJP.v14-604
Mariano Giaquinta and Enrico Giusti, On the regularity of the minima of variational integrals, Acta Math. 148 (1982), 31–46. MR 666107, DOI 10.1007/BF02392725
David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. MR 1814364, DOI 10.1007/978-3-642-61798-0
Moritz Kassmann, A new formulation of Harnack’s inequality for nonlocal operators, C. R. Math. Acad. Sci. Paris 349 (2011), no. 11-12, 637–640 (English, with English and French summaries). MR 2817382, DOI 10.1016/j.crma.2011.04.014
Juha Kinnunen and Tuomo Kuusi, Local behaviour of solutions to doubly nonlinear parabolic equations, Math. Ann. 337 (2007), no. 3, 705–728. MR 2274548, DOI 10.1007/s00208-006-0053-3
Janne Korvenpää, Tuomo Kuusi, and Erik Lindgren, Equivalence of solutions to fractional $p$-Laplace type equations, J. Math. Pures Appl. (9) 132 (2019), 1–26 (English, with English and French summaries). MR 4030247, DOI 10.1016/j.matpur.2017.10.004
Naian Liao, Regularity of weak supersolutions to elliptic and parabolic equations: lower semicontinuity and pointwise behavior, J. Math. Pures Appl. (9) 147 (2021), 179–204 (English, with English and French summaries). MR 4213682, DOI 10.1016/j.matpur.2021.01.008
Peter Lindqvist, Notes on the stationary $p$-Laplace equation, SpringerBriefs in Mathematics, Springer, Cham, 2019. MR 3931688, DOI 10.1007/978-3-030-14501-9
Jan Malý and William P. Ziemer, Fine regularity of solutions of elliptic partial differential equations, Mathematical Surveys and Monographs, vol. 51, American Mathematical Society, Providence, RI, 1997. MR 1461542, DOI 10.1090/surv/051
Neil S. Trudinger, On Harnack type inequalities and their application to quasilinear elliptic equations, Comm. Pure Appl. Math. 20 (1967), 721–747. MR 226198, DOI 10.1002/cpa.3160200406