# $$C^{\sigma +\alpha }$$Cσ+α regularity for concave nonlocal fully nonlinear elliptic equations with rough kernels

@article{Serra2014Csigma,
title={\$\$C^\{\sigma +\alpha \}\$\$C$\sigma$+$\alpha$ regularity for concave nonlocal fully nonlinear elliptic equations with rough kernels},
author={Joaquim Serra},
journal={Calculus of Variations and Partial Differential Equations},
year={2014},
volume={54},
pages={3571-3601}
}
• J. Serra
• Published 5 May 2014
• Mathematics
• Calculus of Variations and Partial Differential Equations
We establish $$C^{\sigma +\alpha }$$Cσ+α interior estimates for concave nonlocal fully nonlinear equations of order $$\sigma \in (0,2)$$σ∈(0,2) with rough kernels. Namely, we prove that if $$u\in C^{\alpha }(\mathbb {R}^n)$$u∈Cα(Rn) solves in $$B_1$$B1 a concave translation invariant equation with kernels in $$\mathcal L_0(\sigma )$$L0(σ), then u belongs to $$C^{\sigma +\alpha }(\overline{B_{1/2}})$$Cσ+α(B1/2¯), with an estimate. More generally, our results allow the equation to depend on x in…
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## References

SHOWING 1-10 OF 15 REFERENCES

We prove space and time regularity for solutions of fully nonlinear parabolic integro-differential equations with rough kernels. We consider parabolic equations $$u_t = \mathrm{I}u$$ut=Iu, where
• Mathematics
• 1997
We study in this chapter a class of partial differential equations that generalize and are to a large extent represented by Laplace’s equation. These are the elliptic partial differential equations
• Mathematics
• 2016
We study fine boundary regularity properties of solutions to fully nonlinear elliptic integro-differential equations of order 2s, with s ∈ (0,1). We consider the class of nonlocal operators L∗ ⊂L 0,
We prove a C 1, α interior regularity theorem for fully nonlinear uniformly elliptic integro-differential equations without assuming any regularity of the kernel. We then give some applications to
• Mathematics
• 2009
We prove a regularity result for solutions of a purely integro-differential Bellman equation. This regularity is enough for the solutions to be understood in the classical sense. If we let the order
• Mathematics
• 2009
We consider nonlinear integro‐differential equations like the ones that arise from stochastic control problems with purely jump Lévy processes. We obtain a nonlocal version of the ABP estimate,
• Mathematics
• 1995
Introduction Preliminaries Viscosity solutions of elliptic equations Alexandroff estimate and maximum principle Harnack inequality Uniqueness of solutions Concave equations $W^{2,p}$ regularity
• Mathematics
• 2009
We obtain C1,α regularity estimates for nonlocal elliptic equations that are not necessarily translation-invariant using compactness and perturbative methods and our previous regularity results for
On etend une theorie de perturbations aux solutions d'equations uniformement elliptiques d'ordre 2 totalement non lineaires