On a mixed Poincare-Steklov type spectral problem in a Lipschitz domain

We consider a mixed boundary value problem for a second-order strongly elliptic equation in a Lipschitz domain. The boundary condition on a part of the boundary is of the first order and contains a weight function and the spectral parameter, while on the remaining part the homogeneous Dirichlet condition is imposed. The aim is to weaken the conditions sufficient for justifying the classical asymptotic formula for the eigenvalues. We show that it suffices to assume the boundary to be C-1 in a neighborhood of the support of the weight outside a closed subset of zero measure.