The structure of the fractional powers of the noncommutative Fourier law

摘要

In the recent years, there has been a lot of interest in fractional diffusion and fractional evolution problems. The spectral theory on the S-spectrum turned out to be an important tool to define new fractional diffusion operators stating from the Fourier law for nonhomogeneous materials. Precisely, let e(l), e(l)=1,2,3 be orthogonal unit vectors in R3 and let Omega subset of R3 be a bounded open set with smooth boundary partial derivative Omega. Denoting by x_ a point in Omega, the heat equation is obtained replacing the Fourier law given by T=ax_partial derivative xe1+bx_partial derivative ye2+cx_partial derivative ze3 into the conservation of energy law. In this paper, we investigate the structure of the fractional powers of the vector operator T, with homogeneous Dirichlet boundary conditions. Recently, we have found sufficient conditions on the coefficients a, b, c:Omega -> R such that the fractional powers of T exist in the sense of the S-spectrum approach. In this paper, we show that under a different set of conditions on the coefficients a, b, c, the fractional powers of T have a different structure.