For any given compact C 2 hypersurface Σ in R 2n bounding a strictly convex set with nonempty interior, in this paper an invariant ̺ n (Σ) is defined and satisfies ̺ n (Σ) ≥ [n/2] + 1, where [a] denotes the greatest integer which is not greater than a ∈ R. The following results are proved in this paper. There always exist at least ̺ n (Σ) geometrically distinct closed characteristics on Σ.

In this paper, we prove that there exist at least two geometrically distinct brake orbits in every bounded convex symmetric domain in R n for n 2.

Abstract. We consider an arbitrary linear elliptic first-order differential operator A with smooth coefficients acting between sections of complex vector bundles E, F over a compact smooth manifold M with smooth boundary Σ. We describe the analytic and topological properties of A in a collar neighborhood U of Σ and analyze various ways of writing A U in product form. We discuss the sectorial projections of the corresponding tangential operator, construct various invertible doubles of A by suitable local boundary conditions, obtain Poisson type operators with different mapping properties, and provide a canonical construction of the Calderón projection. We apply our construction to generalize the Cobordism Theorem and to determine sufficient conditions for continuous variation of the Calderón projection and of well-posed selfadjoint Fredholm extensions under continuous variation of the data.

No abstract

Let Σ be a compact C 2 hypersurface in R 2n bounding a convex set with non-empty interior. In this paper it is proved that there always exist at least n geometrically distinct closed characteristics on Σ if Σ is symmetric with respect to the origin.