Resumen de Exact Green's formula for the fractional Laplacian and perturbations
Let Ω be an open, smooth, bounded subset of Rn. In connection with the fractional Laplacian (−Δ)a (a>0), and more generally for a 2a-order classical pseudodifferential operator (ψdo) P with even symbol, one can define the Dirichlet value γa−10u, resp. Neumann value γa−11u of u(x), as the trace, resp. normal derivative, of u/da−1 on ∂Ω, where d(x) is the distance from x∈Ω to ∂Ω; they define well-posed boundary value problems for P.
A Green's formula was shown in a preceding paper, containing a generally nonlocal term (Bγa−10u,γa−10v)∂Ω, where B is a first-order ψdo on ∂Ω. Presently, we determine B from L in the case P=La, where L is a strongly elliptic second-order differential operator. A particular result is that B=0 when L=−Δ, and that B is multiplication by a function (is local) when L equals −Δ plus a first-order term. In cases of more general L, B can be nonlocal.
