Journal of Integral Equations and Applications

$C^\sigma ,\alpha $ estimates for concave, non-local parabolic equations with critical drift

Héctor Chang Lara and Gonzalo Dávila

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Given a concave integro-differential operator $I$, we study regularity for solutions of fully nonlinear, nonlocal, parabolic equations of the form $u_t-Iu=0$. The kernels are assumed to be smooth but non necessarily symmetric, which accounts for a critical non-local drift. We prove a $C^{\sigma +\alpha }$ estimate in the spatial variable and $C^{1,\alpha }$ estimates in time assuming time regularity for the boundary data. The estimates are uniform in the order of the operator $I$, hence allowing us to extend the classical Evans-Krylov result for concave parabolic equations.

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J. Integral Equations Applications, Volume 28, Number 3 (2016), 373-394.

First available in Project Euclid: 17 October 2016

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Zentralblatt MATH identifier

Primary: 35B45: A priori estimates 35B65: Smoothness and regularity of solutions 35D40: Viscosity solutions 35K55: Nonlinear parabolic equations 35R09: Integro-partial differential equations [See also 45Kxx]

Non-local parabolic equations fully non-linear concave operators Evans-Krylov estimate


Lara, Héctor Chang; Dávila, Gonzalo. $C^\sigma ,\alpha $ estimates for concave, non-local parabolic equations with critical drift. J. Integral Equations Applications 28 (2016), no. 3, 373--394. doi:10.1216/JIE-2016-28-3-373.

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