Exact Green's formula for the fractional Laplacian and perturbations

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Let Omega be an open, smooth, bounded subset of R-n. In connection with the fractional Laplacian (-Delta)(a) (a > 0), and more generally for a 2a-order classical pseudodifferential operator (psi do) P with even symbol, one can define the Dirichlet value gamma(a-1)(0) u, resp. Neumann value gamma(a-1)(1) u of u(x), as the trace, resp. normal derivative, of u/d(a-1) on partial derivative Omega, where d(x) is the distance from x is an element of Omega to partial derivative Omega; they define well-posed boundary value problems for P.

A Green's formula was shown in a preceding paper, containing a generally nonlocal term (B gamma(a-1)(0) u, gamma(a-1)(0) v)partial derivative Omega, where B is a first-order psi do on partial derivative Omega. 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 = -Lambda, and that B is multiplication by a function (is local) when L equals -Delta plus a first-order term. In cases of more general L, B can be nonlocal.

TidsskriftMathematica Scandinavica
Udgave nummer3
Sider (fra-til)568-592
StatusUdgivet - 2020

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