The mixed boundary value problem, Krein resolvent formulas and spectral asymptotic estimates

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:382

Article

Grubb, Gerd¹ 

[1] Univ Copenhagen, Dept Math Sci, DK-2100 Copenhagen, Denmark

关键词: Mixed boundary condition;    Zaremba problem;    Resolvent difference;    Dirichlet-to-Neumann operator;    Krein resolvent formula;    Spectral asymptotics;    Weak Schatten class;    Nonstandard pseudodifferential operator;   

DOI  :  10.1016/j.jmaa.2011.04.055

来源: Elsevier

【 摘 要 】

For a second-order symmetric strongly elliptic operator A on a smooth bounded open set in R-n, the mixed problem is defined by a Neumann-type condition on a part Sigma(+) of the boundary and a Dirichlet condition on the other part Sigma(-). We show a Krein resolvent formula, where the difference between its resolvent and the Dirichlet resolvent is expressed in terms of operators acting on Sobolev spaces over Sigma(+). This is used to obtain a new Weyl-type spectral asymptotics formula for the resolvent difference (where upper estimates were known before), namely s(j) j(2/(n-1)) -> C-0,+(2/(n-1)), where is C-0,C-+ proportional to the area of Sigma(+), in the case where A is principally equal to the Laplacian. (C) 2011 Elsevier Inc. All rights reserved.

【 授权许可】

Free