Operator Estimates for Planar Domains with Irregularly Curved Boundary. The Dirichlet and Neumann Conditions

We consider a general nonselfadjoint second order scalar operator in planar domain such that one of the boundary components is obtained by an arbitrary irregular curving under assumption amplitude small. The Dirichlet or Neumann condition imposed on perturbed component, and remaining part boundary. prove norm resolvent convergence operators, estimate rate two norms, spectrum operators. In selfadjoint case, we establish spectral projections.