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- Institut für Mathematik (9) (remove)

We investigate nonlinear problems which appear as Euler-Lagrange equations for a variational problem. They include in particular variational boundary value problems for nonlinear elliptic equations studied by F. Browder in the 1960s. We establish a solvability criterion of such problems and elaborate an efficient orthogonal projection method for constructing approximate solutions.

The subject of this paper is solutions of an autoresonance equation. We look for a connection between the parameters of the solution bounded as t -> -infinity, and the parameters of two two-parameter families of solutions as t -> infinity. One family consists of the solutions which are not captured into resonance, and another of those increasing solutions which are captured into resonance. In this way we describe the transition through the separatrix for equations with slowly varying parameters and get an estimate for parameters before the resonance of those solutions which may be captured into autoresonance. (C) 2014 AIP Publishing LLC.

The paper is devoted to asymptotic analysis of the Dirichlet problem for a second order partial differential equation containing a small parameter multiplying the highest order derivatives. It corresponds to a small perturbation of a dynamical system having a stationary solution in the domain. We focus on the case where the trajectories of the system go into the domain and the stationary solution is a proper node.

We develop a new approach to the analysis of pseudodifferential operators with small parameter 'epsilon' in (0,1] on a compact smooth manifold X. The standard approach assumes action of operators in Sobolev spaces whose norms depend on 'epsilon'. Instead we consider the cylinder [0,1] x X over X and study pseudodifferential operators on the cylinder which act, by the very nature, on functions depending on 'epsilon' as well. The action in 'epsilon' reduces to multiplication by functions of this variable and does not include any differentiation. As but one result we mention asymptotic of solutions to singular perturbation problems for small values of 'epsilon'.