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For a second-order symmetric strongly elliptic operator A on a smooth boundedopen set \Omega in R^n with boundary \Sigma, the mixed problem is defined by aNeumann-type condition on a part Sigma_+ of the boundary and a Dirichletcondition on the other part Sigma_-. We show a Krein resolvent formula, wherethe difference between its resolvent and the Dirichlet resolvent is expressedin terms of operators acting on Sobolev spaces over Sigma_+. This is used toobtain a new Weyl-type spectral asymptotics formula for the resolventdifference (where upper estimates were known before), namely s_j j^{2/(n-1)}\toC_{0,+}^{2/(n-1)}, where C_{0,+} is proportional to the area of Sigma_+, in thecase where A is principally equal to the Laplacian.

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