Communications in Contemporary Mathematics

This paper addresses the existence of solutions u ∈ H¹(ℝ₊;ℝ^N) of ODE systems , with boundary condition u₁(0) = ξ, where u₁ is a (vector) component of u. Under general conditions, the problem corresponds to a functional equation involving a Fredholm operator with calculable index, which is proper on the closed bounded subsets of H¹(ℝ₊;ℝ^N). When the index is 0 and the solutions are bounded a priori, the existence follows from an available degree theory for such operators. Specific conditions are given that guarantee the existence of a priori bounds and second order equations with Dirichlet, Neumann or initial value conditions are discussed as applications.

This paper is devoted to the study of degenerate critical elliptic equations of Caffarelli–Kohn–Nirenberg type. By means of blow-up analysis techniques, we prove an a priori estimate in a weighted space of continuous functions. From this compactness result, the existence of a solution to our problem is proved by exploiting the homotopy invariance of the Leray–Schauder degree.

A Heisenberg inequality, involving a differentiation operator, its adjoint, and the second quantization operator of a unitary operator, is proved in the context of white noise analysis.

We study complex projective surfaces admitting a Poisson structure; we prove a classification theorem and count how many independent Poisson structures there are on a given Poisson surface.

We study possible extensions of Hardy's inequalities induced by second, fourth and sixth-order linear differential operators with singular potentials.

Let M be a closed even n-manifold of positive sectional curvature on which a torus T^k acts isometrically. We show that if (respectively, k > 1) for n ≠ 12 (respectively, n = 12), then the Euler characteristic of each T^k-fixed point component is positive. This implies that the Euler characteristic of M is positive. We also extend this result to an isometric elementary p-group -action on a closed manifold of positive sectional curvature.

Let Mⁿ be a complete, non-compact and C^∞-smooth Riemannian manifold with non-negative sectional curvature. Suppose that is a soul of Mⁿ given by the fundamental theory of Cheeger and Gromoll, and suppose that is a distance non-increasing retraction from the whole manifold to the soul (e.g. the retraction given by Sharafutdinov). Then we show that the retraction Ψ above must give rise to a C^∞-smooth Riemannian submersion from Mⁿ to the soul . Moreover, we derive a new flat strip theorem associated with the Cheeger–Gromoll convex exhaustion for the manifold above.