zbMATH — the first resource for mathematics

Regularity for solutions of nonlocal, nonsymmetric equations. (English) Zbl 1317.35278
The authors consider integro-differential equations that arise when one studies discontinuous stochastic processes and uses the celebrated Lévy-Khintchine formula. More precisely, the authors study the regularity of solutions of fully nonlinear integro-differential equations with respect to nonsymmetric kernels. They assume that the operator is elliptic with respect to a family of integro-differential linear operators, where the symmetric parts of the kernels have a fixed homogeneity $$\sigma$$ and the skew-symmetric parts have strictly smaller homogeneity $$\tau$$. The authors prove a weak Alexandrov-Bellman-Pucci estimate and $$C^{1,\alpha}$$ regularity. Note that the estimates are uniform as $$\sigma \rightarrow2$$ and $$\tau \rightarrow 1$$, so this result extends the regularity theory for elliptic differential equations with dependence on the gradient.

MSC:
 35R09 Integral partial differential equations 35R60 PDEs with randomness, stochastic partial differential equations 35B65 Smoothness and regularity of solutions to PDEs
Full Text:
References:
 [1] Barles, G.; Imbert, C., Second-order elliptic integro differential equations: viscosity solutions theory revisited, Annales de lʼinstitut Henri Poincaré, analyse non linéaire, 3, 567-585, (2008) · Zbl 1155.45004 [2] Bass, R.F.; Kassmann, M., Hölder continuity of harmonic functions with respect to operators of variable order, Communications in partial differential equations, 8, 1249-1259, (2005) · Zbl 1087.45004 [3] Caffarelli, L.; Cabré, X., Fully nonlinear elliptic equations, American mathematical society colloquium publications, vol. 43, (1995), American Mathematical Society Providence, RI, vi+104 pp · Zbl 0834.35002 [4] Caffarelli, L.; Silvestre, L., Regularity theory for fully nonlinear integro differential equations, Communications on pure and applied mathematics, 5, 597-638, (2009) · Zbl 1170.45006 [5] Caffarelli, L.; Silvestre, L., Regularity results for nonlocal equations by approximation, Archive for rational mechanics and analysis, 1, 59-88, (2011) · Zbl 1231.35284 [6] Caffarelli, L.; Silvestre, L., The Evans-Krylov theorem for non local fully non linear equations, Annals of mathematics, 2, 1163-1187, (2011) · Zbl 1232.49043 [7] Kim, Y.C.; Lee, K.A., Regularity results for fully nonlinear integro differential operators with nonsymmetric positive kernels · Zbl 1258.47064 [8] Silvestre, L., Hölder estimates for solutions of integro-differential equations like the fractional Laplace, Indiana university mathematics journal, 3, 1155-1174, (2006) · Zbl 1101.45004 [9] Soner, H.M., Optimal control with state-space constraint II, SIAM journal on control and optimization, 6, 1110-1122, (1986) · Zbl 0619.49013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.