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 tidsskrift › Tidsskriftartikel › fagfællebedømt
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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