Vol 213, No 4 (2016)

We consider an elliptic operator in a multidimensional domain with frequent alternation of the Dirichlet condition and the Robin boundary condition in the case where the homogenized operator contains only the original Robin boundary condition. We prove the uniform resolvent convergence of the perturbed operator to the homogenized operator and obtain order sharp estimates for the rate of convergence. We construct a complete asymptotic expansion for the resolvent in the case where the resolvent acts on sufficiently smooth functions and the alternation of boundary conditions is strictly periodic and is given on a multidimensional hyperplane. Bibliography: 23 titles.

We consider the boundary value problem for eigenvalues of the negative Laplace operator in a disk with the Neumann boundary condition on the circle except for finitely many (more than 1) small arcs, where the Dirichlet boundary condition is imposed, with lengths tending to zero. We construct complete asymptotics expansions of egenvalues with respect to the parameter (the arc length) converging to a double eigenvalue to the limit Neumann problem, in the critical case, where one of the eigenfunctions of the limit problem vanishes at all contraction points for small arcs.

We study B-potentials of Hölder continuous functions. We obtain formulas for the first– and second order derivatives, as well as for the B-derivative of a B-potential with Hölder density. We also obtain the inversion formula.

The definition of Toeplitz operators in the Bergman space of square integrable analytic functions in the unit disk in the complex plane is extended in such a way that it covers many cases where the traditional definition does not work. This includes, in particular, highly singular symbols such as measures, distributions, and certain hyperfunctions. Bibliography: 22 titles.