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

Abstract

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 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 s j j 2 / ( n − 1 ) → C 0 , + 2 / ( n − 1 ) , where C 0 , + is proportional to the area of Σ + , in the case where A is principally equal to the Laplacian.