Exact Green’s formula for the fractional Laplacian and perturbations - Grubb, Gerd (MATHEMATICA SCANDINAVICA, 2020)
Let Ω be an open, smooth, bounded subset of R^n . 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 γ _0^a-1u, resp. Neumann value γ _1^a-1u of u(x), as the trace, resp. normal derivative, of u/d^a-1 on Ω , where d(x) is the distance from xΩ 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 (Bγ _0^a-1u,γ _0^a-1v)_Ω , where B is a first-order ψdo on Ω . Presently, we determine B from L in the case P=L^a, 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.