We develop a new approach to the analysis of pseudodifferential operators with small parameter epsilon is an element of (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.

This paper considers elliptic problems with high-order derivatives multiplied by a small parameter. We found the algebraic conditions for an operator and the boundary conditions that guarantee the Fredholm property. An a priori estimate for the solution with a constant independent of the small parameter is proved. These results are known for elliptic boundary-value problems with small parameter in the half-space R n + . We extend them to the case of bounded domains with smooth boundary. The small-parameter coercive conditions are formulated, and a two-sided estimate is proved.

We study the Dirichlet problem in a bounded plane domain for the heat equation with small parameter multiplying the derivative in t. The behaviour of solution at characteristic points of the boundary is of special interest. The behaviour is well understood if a characteristic line is tangent to the boundary with contact degree at least 2. We allow the boundary to not only have contact of degree less than 2 with a characteristic line but also a cuspidal singularity at a characteristic point. We construct an asymptotic solution of the problem near the characteristic point to describe how the boundary layer degenerates.