The mixed boundary value problem, Krein resolvent formulas and spectral asymptotic estimates
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The mixed boundary value problem, Krein resolvent formulas and spectral asymptotic estimates. / Grubb, Gerd.
In: Journal of Mathematical Analysis and Applications, Vol. 382, No. 1, 2011, p. 339–363.Research output: Contribution to journal › Journal article › Research › peer-review
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TY - JOUR
T1 - The mixed boundary value problem, Krein resolvent formulas and spectral asymptotic estimates
AU - Grubb, Gerd
PY - 2011
Y1 - 2011
N2 - For a second-order symmetric strongly elliptic operator A on a smooth bounded open set in Rn, the mixed problem is defined by a Neumann-type condition on a part Σ+ of the boundary and a Dirichlet condition on the other part Σ−. We show a Kreĭn resolvent formula, where the difference between its resolvent and the Dirichlet resolvent is expressed in terms of operators acting on Sobolev spaces over Σ+. This is used to obtain a new Weyl-type spectral asymptotics formula for the resolvent difference (where upper estimates were known before), namely sjj2/(n−1)→C0,+2/(n−1), where C0,+ is proportional to the area of Σ+, in the case where A is principally equal to the Laplacian
AB - For a second-order symmetric strongly elliptic operator A on a smooth bounded open set in Rn, the mixed problem is defined by a Neumann-type condition on a part Σ+ of the boundary and a Dirichlet condition on the other part Σ−. We show a Kreĭn resolvent formula, where the difference between its resolvent and the Dirichlet resolvent is expressed in terms of operators acting on Sobolev spaces over Σ+. This is used to obtain a new Weyl-type spectral asymptotics formula for the resolvent difference (where upper estimates were known before), namely sjj2/(n−1)→C0,+2/(n−1), where C0,+ is proportional to the area of Σ+, in the case where A is principally equal to the Laplacian
M3 - Journal article
VL - 382
SP - 339
EP - 363
JO - Journal of Mathematical Analysis and Applications
JF - Journal of Mathematical Analysis and Applications
SN - 0022-247X
IS - 1
ER -
ID: 33793950