On the Estimate of the Absolute Value of Eigenfunctions to the Steklov–Zaremba Problem for the Laplace Operator

The Steklov–Zaremba problem for the Laplace operator in a bounded domain with a strictly Lipschitz boundary is considered. A homogeneous Dirichlet condition is specified on the closed part of the boundary of the domain, and the Steklov boundary condition with a spectral parameter is assumed to be satisfied on the complement to the closed part. This problem is a natural generalization of the classical Steklov problem. With respect to a closed set on the boundary of the domain, where the homogeneous Dirichlet boundary condition is specified, its Wiener capacity is assumed to be positive. It follows from this condition on the capacity that it is natural to consider the problem in the Sobolev space of functions that are square-integrable together with all generalized (weak) first-order derivatives. The aim of the paper is to find an estimate for the maximum modulus of normalized eigenfunctions to the problem under consideration. The proofs of the main results make substantial use of the iterative technique of Jurgen Moser.