Regularity theory for general stable operators
2016; Elsevier BV; Volume: 260; Issue: 12 Linguagem: Inglês
10.1016/j.jde.2016.02.033
ISSN1090-2732
AutoresXavier Ros‐Oton, Joaquim Serra,
Tópico(s)Stochastic processes and financial applications
ResumoWe establish sharp regularity estimates for solutions to Lu=f in Ω⊂Rn, L being the generator of any stable and symmetric Lévy process. Such nonlocal operators L depend on a finite measure on Sn−1, called the spectral measure. First, we study the interior regularity of solutions to Lu=f in B1. We prove that if f is Cα then u belong to Cα+2s whenever α+2s is not an integer. In case f∈L∞, we show that the solution u is C2s when s≠1/2, and C2s−ϵ for all ϵ>0 when s=1/2. Then, we study the boundary regularity of solutions to Lu=f in Ω, u=0 in Rn∖Ω, in C1,1 domains Ω. We show that solutions u satisfy u/ds∈Cs−ϵ(Ω‾) for all ϵ>0, where d is the distance to ∂Ω. Finally, we show that our results are sharp by constructing two counterexamples.
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