Regularity for fully nonlinear nonlocal parabolic equations with rough kernels
2014; Springer Science+Business Media; Volume: 54; Issue: 1 Linguagem: Inglês
10.1007/s00526-014-0798-6
ISSN1432-0835
Autores Tópico(s)Numerical methods in inverse problems
ResumoWe prove space and time regularity for solutions of fully nonlinear parabolic integro-differential equations with rough kernels. We consider parabolic equations $$u_t = \mathrm{I}u$$ , where $$\mathrm{I}$$ is translation invariant and elliptic with respect to the class $$\mathcal L_0(\sigma )$$ of Caffarelli and Silvestre, $$\sigma \in (0,2)$$ being the order of $$\mathrm{I}$$ . We prove that if $$u$$ is a viscosity solution in $$B_1 \times (-1,0]$$ which is merely bounded in $$\mathbb {R}^n \times (-1,0]$$ , then $$u$$ is $$C^\beta $$ in space and $$C^{\beta /\sigma }$$ in time in $$\overline{B_{1/2}} \times [-1/2,0]$$ , for all $$\beta < \min \{\sigma , 1+\alpha \}$$ , where $$\alpha >0$$ . Our proof combines a Liouville type theorem—relaying on the nonlocal parabolic $$C^\alpha $$ estimate of Chang and Dávila—and a blow up and compactness argument.
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