Spectral asymptotics for nonsmooth singular Green operators
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Spectral asymptotics for nonsmooth singular Green operators. / Grubb, Gerd.
I: Communications in Partial Differential Equations, Bind 39, Nr. 3, 2014, s. 530-573.Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt
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TY - JOUR
T1 - Spectral asymptotics for nonsmooth singular Green operators
AU - Grubb, Gerd
PY - 2014
Y1 - 2014
N2 - Singular Green operators G appear typically as boundary correction terms in resolvents for elliptic boundary value problems on a domain Ω ⊂ ℝ n , and more generally they appear in the calculus of pseudodifferential boundary problems. In particular, the boundary term in a Krein resolvent formula is a singular Green operator. It is well-known in smooth cases that when G is of negative order −t on a bounded domain, its eigenvalues ors-numbers have the behavior (*)s j (G) ∼ cj −t/(n−1) for j → ∞, governed by the boundary dimension n − 1. In some nonsmooth cases, upper estimates (**)s j (G) ≤ Cj −t/(n−1) are known.We show that (*) holds when G is a general selfadjoint nonnegative singular Green operator with symbol merely Hölder continuous in x. We also show (*) with t = 2 for the boundary term in the Krein resolvent formula comparing the Dirichlet and a Neumann-type problem for a strongly elliptic second-order differential operator (not necessarily selfadjoint) with coefficients in for some q > n.
AB - Singular Green operators G appear typically as boundary correction terms in resolvents for elliptic boundary value problems on a domain Ω ⊂ ℝ n , and more generally they appear in the calculus of pseudodifferential boundary problems. In particular, the boundary term in a Krein resolvent formula is a singular Green operator. It is well-known in smooth cases that when G is of negative order −t on a bounded domain, its eigenvalues ors-numbers have the behavior (*)s j (G) ∼ cj −t/(n−1) for j → ∞, governed by the boundary dimension n − 1. In some nonsmooth cases, upper estimates (**)s j (G) ≤ Cj −t/(n−1) are known.We show that (*) holds when G is a general selfadjoint nonnegative singular Green operator with symbol merely Hölder continuous in x. We also show (*) with t = 2 for the boundary term in the Krein resolvent formula comparing the Dirichlet and a Neumann-type problem for a strongly elliptic second-order differential operator (not necessarily selfadjoint) with coefficients in for some q > n.
KW - Faculty of Science
KW - Matematik
KW - partielle differentialligninger
U2 - 10.1080/03605302.2013.864207
DO - 10.1080/03605302.2013.864207
M3 - Journal article
VL - 39
SP - 530
EP - 573
JO - Communications in Partial Differential Equations
JF - Communications in Partial Differential Equations
SN - 0360-5302
IS - 3
ER -
ID: 102113374