Acta Mathematica Sinica-English Series最新文献

We establish the well-posedness for a class of McKean–Vlasov SDEs driven by symmetric α-stable Lévy processes (({1 over 2} <alpha leq 1)), where the drift coefficient is Hölder continuous in space variable, while the noise coefficient is Lipscitz continuous in space variable, and both of them satisfy the Lipschitz condition in distribution variable with respect to Wasserstein distance. If the drift coefficient does not depend on distribution variable, our methodology developed in this paper applies to the case α ∈ (0, 1]. The main tool relies on heat kernel estimates for (distribution independent) stable SDEs and Banach’s fixed point theorem.

Let K be a (bounded) closed uniformly convex subset of a Banach space X. We show that

1. (i)

   the nearest point map is well-defined and always continuous from X onto K,

2. (ii)

   there is a reflexive space Y with a uniform rotund in every direction norm such that Y contains K as a subset and the nearest point map P_K: Y → K is uniformly continuous from any bounded set containing K onto K.

We derive a blow-up formula for holomorphic Koszul–Brylinski homologies of compact holomorphic Poisson manifolds. As applications, we investigate the invariance of the E₁-degeneracy of the Dolbeault–Koszul–Brylinski spectral sequence under Poisson blow-ups, and compute the holomorphic Koszul–Brylinski homology for del Pezzo surfaces and two complex nilmanifolds with holomorphic Poisson structures.

In this paper, we study the p-Laplacian equation of the form

on lattice graphs ℤ^N, where N ∈ ℕ*, α ∈ (0, N), (2 leq p < {2Nq over N+alpha}<+infty) and R_α represents the Green’s function of the discrete fractional Laplacian, which has no singularity at the origin but behaves as the Riesz potential at infinity. Under suitable assumptions on the potential h(x), we prove the existence of ground state solutions to the equation above by two different methods.

The aim of this paper is to explore the equivalent characterizations for the boundedness and compactness of C_φ − C_ψ acting from classical (little) Zygmund space ({cal{Z}}({cal{Z}}_{0})) to (little) Bloch-type space ({cal{B}}^{alpha};({cal{B}}_{0}^{alpha})). Especially, we creatively develop a useful lemma, which not only plays a crucial role in the estimations but also offers a sufficient condition for the bounded below property of composition operators.

For each based map f: X → ℝP² from a closed surface into the real projective plane, we compute its absolute and twisted degrees and describe the action of the fundamental group of ℝP² over the based homotopy class of f. We emphasize the finding that for any nonorientable closed surface, there exists only one based homotopy class of maps from it into ℝP² whose maps have twisted degree zero and absolute degree nonzero–which shows that, unlike the absolute degree, the twisted degree is not able to detect the strong surjectivity in this setting. In all the other scenarios, the absolute degree of each map is either equal to the twisted degree or its absolute value–and so the twisted degree detects strong surjectivity.

We prove that for any η that belongs to the closure of the interior of the Markov and Lagrange spectra, the sets k⁻¹((−∞, η]) and k⁻¹(η), which are the sets of irrational numbers with best constant of Diophantine approximation bounded by η and exactly η respectively, have the same Hausdorff dimension. We also show that, as η varies in the interior of the spectra, this Hausdorff dimension is a strictly increasing function.

The aim of this paper is to investigate the existence of solutions to the prescribing fractional Q-curvature problem on ({mathbb S}^{n}) under some reasonable assumption of the Laplacian sign at the critical point of prescribing curvature function K. Due to the lack of compactness, we choose to return to the basic elements of variational theory and study the deformation along the flow lines. The novelty of the paper is that we obtain the existence without assuming any symmetry and periodicity on K. In addition, to overcome the loss of compactness for high-order operator problem, we need more delicate estimates with the second order cases.

In this paper, we give the definition of Maslov-type index of the discrete Hamiltonian system, and obtain the relation of Morse index and Maslov-type index of the discrete Hamiltonian system which is a generalization of the case ω = 1 to ω ∈ U degenerate case via direct method which is different from that of the known literatures. Moreover the well-posedness of the splitting numbers (cal{S}_{h,omega}^{pm}) is proven, then the index iteration theories of Bott and Long are also valid for the discrete case, and those can be also applied to the study of the symplectic algorithm.

In this paper, we consider the Schrödinger type equation −Δu + V (x)u = f(x, u) on the lattice graph ℤ^N with indefinite variational functional, where Δ is the discrete Laplacian. Specifically, we assume that V (x) and f(x, u) are periodic in x, f satisfies some growth condition and 0 lies in a finite spectral gap of (−Δ + V). We obtain ground state solutions by using the method of generalized Nehari manifold which has been introduced by Pankov.