The principal transmission condition

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The principal transmission condition. / Grubb, Gerd.

I: Mathematics In Engineering, Bind 4, Nr. 4, 2022, s. 1-33.

Publikation: Bidrag til tidsskriftTidsskriftartikelfagfællebedømt

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Grubb, G 2022, 'The principal transmission condition', Mathematics In Engineering, bind 4, nr. 4, s. 1-33. https://doi.org/10.3934/mine.2022026

APA

Grubb, G. (2022). The principal transmission condition. Mathematics In Engineering, 4(4), 1-33. https://doi.org/10.3934/mine.2022026

Vancouver

Grubb G. The principal transmission condition. Mathematics In Engineering. 2022;4(4):1-33. https://doi.org/10.3934/mine.2022026

Author

Grubb, Gerd. / The principal transmission condition. I: Mathematics In Engineering. 2022 ; Bind 4, Nr. 4. s. 1-33.

Bibtex

@article{755afe02a4394d119152b405494d8afe,
title = "The principal transmission condition",
abstract = "The paper treats pseudodifferential operators P=Op(p(ξ)) with homogeneous complex symbol p(ξ) of order 2a>0, generalizing the fractional Laplacian (−Δ)a but lacking its symmetries, and taken to act on the halfspace Rn+. The operators are seen to satisfy a principal μ-transmission condition relative to Rn+, but generally not the full μ-transmission condition satisfied by (−Δ)a and related operators (with μ=a). However, P acts well on the so-called μ-transmission spaces over Rn+ (defined in earlier works), and when P moreover is strongly elliptic, these spaces are the solution spaces for the homogeneous Dirichlet problem for P, leading to regularity results with a factor xμn (in a limited range of Sobolev spaces). The information is then shown to be sufficient to establish an integration by parts formula over Rn+ for P acting on such functions. The formulation in Sobolev spaces, and the results on strongly elliptic operators going beyond certain operators with real kernels, are new. Furthermore, large solutions with nonzero Dirichlet traces are described, and a halfways Green's formula is established, as new results for these operators. Since the principal μ-transmission condition has weaker requirements than the full μ-transmission condition assumed in earlier papers, new arguments were needed, relying on work of Vishik and Eskin instead of the Boutet de Monvel theory. The results cover the case of nonsymmetric operators with real kernel that were only partially treated in a preceding paper.",
keywords = "fractional-order pseudodi fferential operator, alpha-stable L'evy process, homogeneous symbol, Dirichlet problem on the halfspace, regularity estimate, halfways Green's formula",
author = "Gerd Grubb",
year = "2022",
doi = "10.3934/mine.2022026",
language = "English",
volume = "4",
pages = "1--33",
journal = "Mathematics In Engineering",
issn = "2640-3501",
publisher = "American Institute of Mathematical Sciences",
number = "4",

}

RIS

TY - JOUR

T1 - The principal transmission condition

AU - Grubb, Gerd

PY - 2022

Y1 - 2022

N2 - The paper treats pseudodifferential operators P=Op(p(ξ)) with homogeneous complex symbol p(ξ) of order 2a>0, generalizing the fractional Laplacian (−Δ)a but lacking its symmetries, and taken to act on the halfspace Rn+. The operators are seen to satisfy a principal μ-transmission condition relative to Rn+, but generally not the full μ-transmission condition satisfied by (−Δ)a and related operators (with μ=a). However, P acts well on the so-called μ-transmission spaces over Rn+ (defined in earlier works), and when P moreover is strongly elliptic, these spaces are the solution spaces for the homogeneous Dirichlet problem for P, leading to regularity results with a factor xμn (in a limited range of Sobolev spaces). The information is then shown to be sufficient to establish an integration by parts formula over Rn+ for P acting on such functions. The formulation in Sobolev spaces, and the results on strongly elliptic operators going beyond certain operators with real kernels, are new. Furthermore, large solutions with nonzero Dirichlet traces are described, and a halfways Green's formula is established, as new results for these operators. Since the principal μ-transmission condition has weaker requirements than the full μ-transmission condition assumed in earlier papers, new arguments were needed, relying on work of Vishik and Eskin instead of the Boutet de Monvel theory. The results cover the case of nonsymmetric operators with real kernel that were only partially treated in a preceding paper.

AB - The paper treats pseudodifferential operators P=Op(p(ξ)) with homogeneous complex symbol p(ξ) of order 2a>0, generalizing the fractional Laplacian (−Δ)a but lacking its symmetries, and taken to act on the halfspace Rn+. The operators are seen to satisfy a principal μ-transmission condition relative to Rn+, but generally not the full μ-transmission condition satisfied by (−Δ)a and related operators (with μ=a). However, P acts well on the so-called μ-transmission spaces over Rn+ (defined in earlier works), and when P moreover is strongly elliptic, these spaces are the solution spaces for the homogeneous Dirichlet problem for P, leading to regularity results with a factor xμn (in a limited range of Sobolev spaces). The information is then shown to be sufficient to establish an integration by parts formula over Rn+ for P acting on such functions. The formulation in Sobolev spaces, and the results on strongly elliptic operators going beyond certain operators with real kernels, are new. Furthermore, large solutions with nonzero Dirichlet traces are described, and a halfways Green's formula is established, as new results for these operators. Since the principal μ-transmission condition has weaker requirements than the full μ-transmission condition assumed in earlier papers, new arguments were needed, relying on work of Vishik and Eskin instead of the Boutet de Monvel theory. The results cover the case of nonsymmetric operators with real kernel that were only partially treated in a preceding paper.

KW - fractional-order pseudodi fferential operator

KW - alpha-stable L'evy process

KW - homogeneous symbol

KW - Dirichlet problem on the halfspace

KW - regularity estimate

KW - halfways Green's formula

U2 - 10.3934/mine.2022026

DO - 10.3934/mine.2022026

M3 - Journal article

VL - 4

SP - 1

EP - 33

JO - Mathematics In Engineering

JF - Mathematics In Engineering

SN - 2640-3501

IS - 4

ER -

ID: 284408334