Exact Green's formula for the fractional Laplacian and perturbations
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Exact Green's formula for the fractional Laplacian and perturbations. / Grubb, Gerd.
I: Mathematica Scandinavica, Bind 126, Nr. 3, 2020, s. 568-592.Publikation: Bidrag til tidsskrift › Tidsskriftartikel › fagfællebedømt
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TY - JOUR
T1 - Exact Green's formula for the fractional Laplacian and perturbations
AU - Grubb, Gerd
PY - 2020
Y1 - 2020
N2 - 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.
AB - 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.
KW - MU-TRANSMISSION
U2 - 10.7146/math.scand.a-120889
DO - 10.7146/math.scand.a-120889
M3 - Journal article
VL - 126
SP - 568
EP - 592
JO - Mathematica Scandinavica
JF - Mathematica Scandinavica
SN - 0025-5521
IS - 3
ER -
ID: 257707144