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

For a second-order symmetric strongly elliptic operator A on a smooth bounded open set \Omega in R^n with boundary \Sigma, 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)}\to C_{0,+}^{2/(n-1)}, where C_{0,+} is proportional to the area of Sigma_+, in the case where A is principally equal to the Laplacian.