Leonid Volevich

Boundary value problems for a class of elliptic operator pencils.

In this paper operator pencilsA(x, D, λ) are studied which act on a manifold with boundary and satisfy the condition of N-ellipticity with parameter, a generalization of the notion of ellipticity with parameter as introduced by Agmon and Agranovich-Vishik. Sobolev spaces corresponding to the Newton polygon are defined and investigated; in particular it is possible to describe their trace spaces. With respect to these spaces, an a priori estimate is proved for the Dirichlet boundary value problem connected with an N-elliptic pencil.

On elliptic operator pencils with general boundary conditions

In this paper parameter-dependent partial differential operators are investigated which satisfy the condition of N-ellipticity with parameter, an ellipticity condition formulated with the use of the Newton polygon. For boundary value problems with general boundary operators we define N-ellipticity including an analogue of the Shapiro-Lopatinskii condition. It is show that the boundary value problem is N-elliptic if and only if an a priori estimate with respect to certain parameter-dependent norms holds. These results are closely connected with singular perturbation theory and lead to uniform estimates, for problems of Vishik-Lyusternik type containing a small parameter.

Exponential Dichotomy and Time-Bounded Solutions for First-Order Hyperbolic Systems

The paper is devoted to studying a class of strongly hyperbolic systems of the first order. We show that if the characteristic roots of the full symbol are outside an open strip containing the real axis, then the homogeneous system possesses an exponential dichotomy and the inhomogeneous system is solvable in the space of time-bounded and almost periodic functions. We also discuss some results on the behavior of solutions for nonlinear equations in the neighborhood of a stationary point.

Parameter-Elliptic Boundary Value Problems and their Formal Asymptotic Solutions

We consider boundary value problems for mixed-order systems of partial differential operators which depend on a complex parameter but which are not parameter-elliptic in the sense of Agmon and Agranovich-Vishik. Such systems are closely related to the theory of singularly perturbed problems. Under the condition of so-called weak parameter-ellipticity it is possible to construct the formal asymptotic solution which shows, in particular, the existence of boundary layers.