• DOI:10.1007/S00526-015-0914-2
• Corpus ID: 73651477

$$C^{\sigma +\alpha }$$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 a $$C^\alpha $$Cα fashion. Our method of proof combines a Liouville theorem and a blow-up (compactness) procedure. Due to its flexibility, the same method can be useful in different regularity proofs for nonlocal equations.