Spectral results for mixed problems and fractional elliptic operators,
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Spectral results for mixed problems and fractional elliptic operators, / Grubb, Gerd.
I: Journal of Mathematical Analysis and Applications, Bind 421 , Nr. 2, 2015, s. 1616-1634.Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt
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TY - JOUR
T1 - Spectral results for mixed problems and fractional elliptic operators,
AU - Grubb, Gerd
PY - 2015
Y1 - 2015
N2 - In the first part of the paper we show Weyl type spectral asymptotic formulas for pseudodifferential operators P a of order 2a, with type and factorization index a ∈ R+, restricted to compact sets with boundary; this includes fractional powers of the Laplace operator. The domain and the regularity of eigenfunctions is described. In the second part, we apply this in a study of realizations Aχ,Σ+ in L2(Ω) of mixed problems for a second-order strongly elliptic symmetric differential operator A on a bounded smooth set Ω ⊂ Rn; here the boundary ∂Ω=Σ is partioned smoothly into Σ=Σ_∪Σ+, the Dirichlet condition γ0u=0 is imposed on Σ_, and a Neumann or Robin condition χu=0 is imposed on Σ+. It is shown that the Dirichlet-to-Neumann operator Pγ,χ is principally of type 1/2 with factorization index 1/2, relative to Σ+. The above theory allows a detailed description of D (Aχ,Σ_+) with singular elements outside of Η3/2 (Ω), and leads to a spectral asymptotic formula for the Krein resolvent difference A −1χ,Σ_+ − A−1ϒ.
AB - In the first part of the paper we show Weyl type spectral asymptotic formulas for pseudodifferential operators P a of order 2a, with type and factorization index a ∈ R+, restricted to compact sets with boundary; this includes fractional powers of the Laplace operator. The domain and the regularity of eigenfunctions is described. In the second part, we apply this in a study of realizations Aχ,Σ+ in L2(Ω) of mixed problems for a second-order strongly elliptic symmetric differential operator A on a bounded smooth set Ω ⊂ Rn; here the boundary ∂Ω=Σ is partioned smoothly into Σ=Σ_∪Σ+, the Dirichlet condition γ0u=0 is imposed on Σ_, and a Neumann or Robin condition χu=0 is imposed on Σ+. It is shown that the Dirichlet-to-Neumann operator Pγ,χ is principally of type 1/2 with factorization index 1/2, relative to Σ+. The above theory allows a detailed description of D (Aχ,Σ_+) with singular elements outside of Η3/2 (Ω), and leads to a spectral asymptotic formula for the Krein resolvent difference A −1χ,Σ_+ − A−1ϒ.
KW - Faculty of Science
KW - matematik
U2 - 10.1016/j.jmaa.2014.07.081
DO - 10.1016/j.jmaa.2014.07.081
M3 - Journal article
VL - 421
SP - 1616
EP - 1634
JO - Journal of Mathematical Analysis and Applications
JF - Journal of Mathematical Analysis and Applications
SN - 0022-247X
IS - 2
ER -
ID: 126422901