Fractional powers of vector operators with first order boundary conditions

Recently the S-spectrum approach to fractional diffusion problems has been applied to vector operators with homogeneous Dirichlet boundary conditions. This method allows to determine the Fractional Fourier law under given boundary conditions. In this paper we consider first order linear boundary conditions and we study the case of vector operators with commuting components. Precisely, let Ω be an open bounded set in R3 where its boundary ∂Ω is considered suitably regular, we denote a point in Ω¯ by x=(x1,x2,x3) and a basis for the quaternions H will be indicated by eℓ, for ℓ=1,2,3. We prove that under suitable conditions on the coefficients a1, a2, a3:Ω¯⊂R3→R of the vector operator T=e1a1(x1)∂xjavax.xml.bind.JAXBElement@1faa5836+e2a2(x2)∂xjavax.xml.bind.JAXBElement@730b0813+e3a3(x3)∂xjavax.xml.bind.JAXBElement@30847b7c,x∈Ω¯ and on the coefficient a:∂Ω→R of the boundary operator B: B≔∑ℓ=13aℓ2(xℓ)nℓ∂xjavax.xml.bind.JAXBElement@1c509deb+a(x)I,x∈∂Ω, where n=(n1,n2,n3) is the outward unit normal vector to ∂Ω, we can define the fractional powers Tα, for α∈(0,1), of T. In general the coefficients a1, a2, a3 and a can depend on time. We omit the time dependence for the sake of simplicity but the proofs of our results can be easily extended to this more general setting considering the time as a parameter.