Regularity for solutions of nonlocal, nonsymmetric equations
Annales de l'I.H.P. Analyse non linéaire, Volume 29 (2012) no. 6, p. 833-859

We study the regularity for solutions of fully nonlinear integro differential equations with respect to nonsymmetric kernels. More precisely, we assume that our operator is elliptic with respect to a family of integro differential linear operators where the symmetric parts of the kernels have a fixed homogeneity σ and the skew symmetric parts have strictly smaller homogeneity τ. We prove a weak ABP estimate and ${C}^{1,\alpha }$ regularity. Our estimates remain uniform as we take $\sigma \to 2$ and $\tau \to 1$ so that this extends the regularity theory for elliptic differential equations with dependence on the gradient.

@article{AIHPC_2012__29_6_833_0,
author = {Chang Lara, H\'ector and D\'avila, Gonzalo},
title = {Regularity for solutions of nonlocal, nonsymmetric equations},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
publisher = {Elsevier},
volume = {29},
number = {6},
year = {2012},
pages = {833-859},
doi = {10.1016/j.anihpc.2012.04.006},
zbl = {1317.35278},
mrnumber = {2995098},
language = {en},
url = {http://www.numdam.org/item/AIHPC_2012__29_6_833_0}
}

Chang Lara, Héctor; Dávila, Gonzalo. Regularity for solutions of nonlocal, nonsymmetric equations. Annales de l'I.H.P. Analyse non linéaire, Volume 29 (2012) no. 6, pp. 833-859. doi : 10.1016/j.anihpc.2012.04.006. http://www.numdam.org/item/AIHPC_2012__29_6_833_0/

[1] G. Barles, C. Imbert, Second-order elliptic integro differential equations: viscosity solutions theory revisited, Annales de lʼInstitut Henri Poincaré, Analyse Non Linéaire 3 (2008), 567-585 | Numdam | MR 2422079 | Zbl 1155.45004

[2] R.F. Bass, M. Kassmann, Hölder continuity of harmonic functions with respect to operators of variable order, Communications in Partial Differential Equations 8 (2005), 1249-1259 | MR 2180302 | Zbl 1087.45004

[3] L. Caffarelli, X. Cabré, Fully Nonlinear Elliptic Equations, American Mathematical Society Colloquium Publications vol. 43, American Mathematical Society, Providence, RI (1995) | MR 1351007 | Zbl 0834.35002

[4] L. Caffarelli, L. Silvestre, Regularity theory for fully nonlinear integro differential equations, Communications on Pure and Applied Mathematics 5 (2009), 597-638 | MR 2494809 | Zbl 1170.45006

[5] L. Caffarelli, L. Silvestre, Regularity results for nonlocal equations by approximation, Archive for Rational Mechanics and Analysis 1 (2011), 59-88 | MR 2781586 | Zbl 1231.35284

[6] L. Caffarelli, L. Silvestre, The Evans–Krylov theorem for non local fully non linear equations, Annals of Mathematics 2 (2011), 1163-1187 | MR 2831115 | Zbl 1232.49043

[7] Y.C. Kim, K.A. Lee, Regularity results for fully nonlinear integro differential operators with nonsymmetric positive kernels, arXiv:1011.3565v2 [math.AP] | MR 2974279 | Zbl 1258.47064

[8] L. Silvestre, Hölder estimates for solutions of integro-differential equations like the fractional Laplace, Indiana University Mathematics Journal 3 (2006), 1155-1174 | MR 2244602 | Zbl 1101.45004

[9] H.M. Soner, Optimal control with state-space constraint II, SIAM Journal on Control and Optimization 6 (1986), 1110-1122 | MR 861089 | Zbl 0619.49013