Smooth approximations for fully nonlinear nonlocal elliptic equations

Fernández-Real, Xavier

doi:10.1090/tran/9038

Smooth approximations for fully nonlinear nonlocal elliptic equations
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by Xavier Fernández-Real;

Trans. Amer. Math. Soc. 377 (2024), 495-515

DOI: https://doi.org/10.1090/tran/9038

Published electronically: October 19, 2023

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Abstract:

We show that any viscosity solution to a general fully nonlinear nonlocal elliptic equation can be approximated by smooth ($C^\infty$) solutions.

References

Xavier Cabré, Serena Dipierro, and Enrico Valdinoci, The Bernstein technique for integro-differential equations, Arch. Ration. Mech. Anal. 243 (2022), no. 3, 1597–1652. MR 4381148, DOI 10.1007/s00205-021-01749-x
Luis Caffarelli and Luis Silvestre, Regularity theory for fully nonlinear integro-differential equations, Comm. Pure Appl. Math. 62 (2009), no. 5, 597–638. MR 2494809, DOI 10.1002/cpa.20274
Luis Caffarelli and Luis Silvestre, Smooth approximations of solutions to nonconvex fully nonlinear elliptic equations, Nonlinear partial differential equations and related topics, Amer. Math. Soc. Transl. Ser. 2, vol. 229, Amer. Math. Soc., Providence, RI, 2010, pp. 67–85. MR 2667633, DOI 10.1090/trans2/229/05
Luis Caffarelli and Luis Silvestre, The Evans-Krylov theorem for nonlocal fully nonlinear equations, Ann. of Math. (2) 174 (2011), no. 2, 1163–1187. MR 2831115, DOI 10.4007/annals.2011.174.2.9
Luis Caffarelli and Luis Silvestre, Regularity results for nonlocal equations by approximation, Arch. Ration. Mech. Anal. 200 (2011), no. 1, 59–88. MR 2781586, DOI 10.1007/s00205-010-0336-4
Luis A. Caffarelli and Panagiotis E. Souganidis, A rate of convergence for monotone finite difference approximations to fully nonlinear, uniformly elliptic PDEs, Comm. Pure Appl. Math. 61 (2008), no. 1, 1–17. MR 2361302, DOI 10.1002/cpa.20208
Luis A. Caffarelli and Panagiotis E. Souganidis, Rates of convergence for the homogenization of fully nonlinear uniformly elliptic pde in random media, Invent. Math. 180 (2010), no. 2, 301–360. MR 2609244, DOI 10.1007/s00222-009-0230-6
Serena Dipierro, Xavier Ros-Oton, Joaquim Serra, and Enrico Valdinoci, Non-symmetric stable operators: regularity theory and integration by parts, Adv. Math. 401 (2022), Paper No. 108321, 100. MR 4395952, DOI 10.1016/j.aim.2022.108321
Lawrence C. Evans, Partial differential equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. MR 1625845, DOI 10.1090/gsm/019
Xavier Fernández-Real and Xavier Ros-Oton, Regularity theory for elliptic PDE, Zurich Lectures in Advanced Mathematics, vol. 28, EMS Press, Berlin, [2022] ©2022. MR 4560756, DOI 10.4171/zlam/28
X. Fernández-Real and X. Ros-Oton, Integro-Differential Elliptic Equations, Progress in Mathematics, Birkhäuser Cham, 2024.
David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Grundlehren der Mathematischen Wissenschaften, Vol. 224, Springer-Verlag, Berlin-New York, 1977. MR 473443, DOI 10.1007/978-3-642-96379-7
Dennis Kriventsov, $C^{1,\alpha }$ interior regularity for nonlinear nonlocal elliptic equations with rough kernels, Comm. Partial Differential Equations 38 (2013), no. 12, 2081–2106. MR 3169771, DOI 10.1080/03605302.2013.831990
Chenchen Mou, Existence of $C^\alpha$ solutions to integro-PDEs, Calc. Var. Partial Differential Equations 58 (2019), no. 4, Paper No. 143, 28. MR 3989947, DOI 10.1007/s00526-019-1597-x
Xavier Ros-Oton and Joaquim Serra, Regularity theory for general stable operators, J. Differential Equations 260 (2016), no. 12, 8675–8715. MR 3482695, DOI 10.1016/j.jde.2016.02.033
X. Ros-Oton, C. Torres-Latorre, and M. Weidner, Semiconvexity estimates for nonlinear integro-differential equations, preprint arXiv (2023).
Joaquim Serra, $C^{\sigma +\alpha }$ regularity for concave nonlocal fully nonlinear elliptic equations with rough kernels, Calc. Var. Partial Differential Equations 54 (2015), no. 4, 3571–3601. MR 3426087, DOI 10.1007/s00526-015-0914-2
Luis Silvestre, Hölder estimates for solutions of integro-differential equations like the fractional Laplace, Indiana Univ. Math. J. 55 (2006), no. 3, 1155–1174. MR 2244602, DOI 10.1512/iumj.2006.55.2706
Hui Yu, A Dirichlet problem for nonlocal degenerate elliptic operators with internal nonlinearity, J. Math. Anal. Appl. 448 (2017), no. 1, 326–346. MR 3579887, DOI 10.1016/j.jmaa.2016.11.005
Hui Yu, Smooth solutions to a class of nonlocal fully nonlinear elliptic equations, Indiana Univ. Math. J. 66 (2017), no. 6, 1895–1919. MR 3744813, DOI 10.1512/iumj.2017.66.6218