Multidimensional Dirichlet series in the problem of the asymptotics of spectral series of nonlinear elliptic operators

Abstract

Series of asymptotic solutions of nonlinear elliptic boundary-value problems in compact domains with a spectral parameter contained in the boundary condition are constructed, and the connection of these solutions with the trajectories of classical Hamiltonian systems defined on the boundary of the domains considered is established. The asymptotic solutions indicated are expressed in terms of multidimensional Dirichlet series, and a “superposition law” is established for them which, as it turns out, does not depend either on the number of independent variables in the original problem or on the form of the nonlinearity.