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 -