Related papers: Sharp convergence for sequences of nonelliptic Sch…

We consider the metric transformation of metric measure spaces/pyramids. We clarify the conditions to obtain the convergence of the sequence of transformed spaces from that of the original sequence, and, conversely, to obtain the…

We consider the Calogero-Sutherland derivative nonlinear Schr\"odinger equation in the focusing (with sign $+$) and defocusing case (with sign $-$) $$ i\partial_tu+\partial_x^2u\,\pm\,\frac2i\,\partial_x\Pi(|u|^2)u=0\,,\qquad…

Motivated by a recent work of Chen-Zheng [8] on Strominger space forms, we prove that a compact Hermitian surface with pointwise constant holomorphic sectional curvature with respect to a Gauduchon connection $\nabla^t $ is either K\"ahler,…

We establish multiple recurrence and convergence results for pairs of zero entropy measure preserving transformations that do not satisfy any commutativity assumptions. Our results cover the case where the iterates of the two…

Let $(M^n,g)$ be a Riemannian manifold without boundary. We study the amount of initial regularity is required so that the solution to free Schr\"{o}dinger equation converges pointwisely to its initial data. Assume the initial data is in…

We consider a fractal refinement of the Carleson problem for the Schr\"odinger equation, that is to identify the minimal regularity needed by the solutions to converge pointwise to their initial data almost everywhere with respect to the…

The aim of this paper is to establish strong convergence theorems for a strongly relatively nonexpansive sequence in a smooth and uniformly convex Banach space. Then we employ our results to approximate solutions of the zero point problem…

Second-order two-scale expansions, a unified proof for the regularity of the correctors based on the translation invariant and a lemma for extracting $O(\epsilon)$ from the remainder term are presented for the second order nonlinear…

This paper is dedicated to studying pointwise estimates of the fundamental solution for the higher order Schr\"{o}dinger equation: % we investigate the fundamental solution of the higher order Schr\"{o}dinger equation…

In this paper, we consider global solutions of the following nonlinear Schr\"odinger equation $iu_t+\Delta u+\lambda|u|^\alpha u = 0,$ in $\R^N,$ with $\lambda\in\R,$ $\alpha\in(0,\frac{4}{N-2})$ $(\alpha\in(0,\infty)$ if $N=1)$ and…

We consider non-gauge-invariant cubic nonlinear Schr\"odinger equations in one space dimension. We show that initial data of size $\varepsilon$ in a weighted Sobolev space lead to solutions with sharp $L_x^\infty$ decay up to time…

We shall generalize the concept of $z=(1-t)x\oplus ty$ to $n$ times which contains to verifying some their properties and inequalities in CAT(0) spaces. In the sequel with introducing of $\alpha$-nonexpansive mappings, we obtain some fixed…

We show that a vector-valued Kahn--Kalai--Linial inequality holds in every Banach space of Rademacher type 2. We also show that for any nondecreasing function $h\geq 0$ with $0<\int_{1}^{\infty}\frac{h(t)}{t^{2}}\mathrm{dt}<\infty$ we have…

We show that the positive mass theorem holds for continuous Riemannian metrics that lie in the Sobolev space $W^{2, n/2}_{loc}$ for manifolds of dimension less than or equal to $7$ or spin-manifolds of any dimension. More generally, we give…

We consider equations of nonlinear Schrodinger type augmented by nonlinear damping terms. We show that nonlinear damping prevents finite time blow-up in several situations, which we describe. We also prove that the presence of a quadratic…

We prove that solutions of the toroidal Schr{\"o}dinger equation can be observed from suitably curved space-time trajectories, thus of zero Lebesgue measure. To do so, we establish new upper and lower bounds for certain trigonometric sums…

We consider non-expanding shear free (NE-SF) null surface geometries embeddable as extremal Killing horizons to the second order in Einstein vacuum spacetimes. A NE-SF null surface geometry consists of a degenerate metric tensor and a…

We prove existence of a special class of solutions to the (elliptic) Nonlinear Schroeodinger Equation $- \epsilon^2 \Delta \psi + V(x) \psi = |\psi|^{p-1} \psi$, on a manifold or in the Euclidean space. Here V represents the potential, p an…

We show that an $n$-dimensional surface whose entropy is close to that of an $n$-dimensional plane is close in Hausdorff distance to some $n$-dimensional plane at every scale. Moreover we show that self-expanders of low entropy converge in…

We study both function theoretic and spectral properties on complete noncompact smooth metric measure space $(M,g,e^{-f}dv)$ with nonnegative Bakry-\'{E}mery Ricci curvature. Among other things, we derive a gradient estimate for positive…