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 tidsskriftTidsskriftartikelfagfællebedømt

Harvard

Grubb, G 2020, 'Exact Green's formula for the fractional Laplacian and perturbations', Mathematica Scandinavica, bind 126, nr. 3, s. 568-592. https://doi.org/10.7146/math.scand.a-120889

APA

Grubb, G. (2020). Exact Green's formula for the fractional Laplacian and perturbations. Mathematica Scandinavica, 126(3), 568-592. https://doi.org/10.7146/math.scand.a-120889

Vancouver

Grubb G. Exact Green's formula for the fractional Laplacian and perturbations. Mathematica Scandinavica. 2020;126(3):568-592. https://doi.org/10.7146/math.scand.a-120889

Author

Grubb, Gerd. / Exact Green's formula for the fractional Laplacian and perturbations. I: Mathematica Scandinavica. 2020 ; Bind 126, Nr. 3. s. 568-592.

Bibtex

@article{8fa7e78fa85f4b61a4b2a4f90f1aa764,
title = "Exact Green's formula for the fractional Laplacian and perturbations",
abstract = "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.",
keywords = "MU-TRANSMISSION",
author = "Gerd Grubb",
year = "2020",
doi = "10.7146/math.scand.a-120889",
language = "English",
volume = "126",
pages = "568--592",
journal = "Mathematica Scandinavica",
issn = "0025-5521",
publisher = "Aarhus Universitet * Mathematica Scandinavica",
number = "3",

}

RIS

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