Fractional elliptic equations, Caccioppoli estimates and regularity

Let L=−divx(A(x)∇x)L=−divx(A(x)∇x) be a uniformly elliptic operator in divergence form in a bounded domain Ω. We consider the fractional nonlocal equations{Lsu=f,in Ω,u=0,on ∂Ω,and{Lsu=f,in Ω,∂Au=0,on ∂Ω. Here LsLs, 0