Abstract
In this paper, we consider fully nonlinear integro-differential equations with possibly nonsymmetric kernels. We are able to find different versions of Alexandroff–Backelman–Pucci estimate corresponding to the full class \({\mathcal {S}^{\mathfrak {L}_0}}\) of uniformly elliptic nonlinear equations with 1 < σ < 2 (subcritical case) and to their subclass \({\mathcal {S}_{\eta}^{\mathfrak {L}_0}}\) with 0 < σ ≤ 1. We show that \({\mathcal {S}_{\eta}^{\mathfrak {L}_0}}\) still includes a large number of nonlinear operators as well as linear operators. And we show a Harnack inequality, Hölder regularity, and C 1,α-regularity of the solutions by obtaining decay estimates of their level sets in each cases.
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References
Barlow M.T., Bass R.F., Chen Z.-Q., Kassmann M.: Non-local Dirichlet forms and symmetric jump processes. Trans. Am. Math. Soc 361(4), 1963–1999 (2009)
Bass R.F., Kassmann M.: Harnack inequalities for non-local operators of variable order. Trans. Am. Math. Soc 357(2), 837–850 (2005)
Bass R.F., Kassmann M.: Hölder continuity of harmonic functions with respect to operators of variable order. Commun. Partial Differ. Equ 30(7–9), 1249–1259 (2005)
Bass R.F., Levin D.A.: Harnack inequalities for jump processes. Potential Anal 17(4), 375–388 (2002)
Caffarelli, L.A., Cabré, X.: Fully Nonlinear Elliptic Equations, vol. 43 of American Mathematical Society, Colloquium Publications. American Mathematical Society, Providence, RI (1995)
Caffarelli L.A., Silvestre L.: Regularity theory for fully nonlinear integro-differential equations. Commun. Pure Appl. Math 62(5), 597–638 (2009)
Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, 2nd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224. Springer, Berlin (1983)
Jensen R.: The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations. Arch. Rational Mech. Anal. 101(1), 1–27 (1988)
Kim,Y.-C., Lee, K.-A.: Regularity results for fully nonlinear integro-differential operators with nonsymmetric positive kernels. Subcritical Case, submitted. http://arxiv.org/abs/1006.0608
Krylov N.V., Safonov M.V.: An estimate for the probability of a diffusion process hitting a set of positive measure. Dokl. Akad. Nauk. SSSR 245, 18–20 (1979)
Lara, H.C., Dávila, G.: Regularity for Solutions of Nonlocal, Nonsymmetric Equations. http://arxiv.org/abs/1106.6070
Silvestre L.: Hölder estimates for solutions of integro-differential equations like the fractional laplace. Indiana Univ. Math. J 55(3), 1155–1174 (2006)
Song R., Vondraček Z.: Harnack inequality for some classes of Markov processes. Math. Z 246(1–2), 177–202 (2004)
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Kim, YC., Lee, KA. Regularity results for fully nonlinear integro-differential operators with nonsymmetric positive kernels. manuscripta math. 139, 291–319 (2012). https://doi.org/10.1007/s00229-011-0516-z
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DOI: https://doi.org/10.1007/s00229-011-0516-z