Weighted Strichartz Estimates for the Radial Perturbed Schrödinger Equation on the Hyperbolic Space

In this paper, we study the radial Schrödinger equation perturbed with a rough time dependent potential on the hyperbolic space. It is natural to expect that the curvature of the manifold has some influence on the dispersive properties, indeed we obtain the weighted Strichartz estimates for the perturbed Cauchy problem. We shall notice that our weighted Strichartz estimates makes possible to treat the nonlinearity of the form g(Ω, u) which are unbounded as |Ω| → ∞.     

On the isolated singularities of the solutions of the Gaussian curvature equation for nonnegative curvature

The precise asymptotic behaviour of the solutions to the two-dimensional curvature equation Δu=k(z)e2u with e2u∈L¹ for bounded nonnegative curvature functions −k(z) near isolated singularities is obtained.     

Quantitative unique continuation for Neumann problems of elliptic equations with weight

Let P(ω, ?) be an elliptic operator with weight ω, and let u be a solution in some Lipschitz domains to ?P(ω, ?u)+W?u+Vu=0 with sharp singular potentials W and V. The weighted doubling estimates, the weighted three‐ball inequalities and the unique continuation at the boundary for solution u of the Neumann problem are established in this paper. Copyright © 2009 John Wiley & Sons, Ltd.     

Application of Andrews and Green-Osher inequalities to nonlocal flow of convex plane curves

We shall apply certain inequalities by Andrews and Green-Osher to area-preserving and length-preserving flows of convex plane curves and show that, if we have bound on the curvature, the evolving curves will converge to a round circle.     

On the curve diffusion flow of closed plane curves

In this paper, we consider the steepest descent H ^?1-gradient flow of the length functional for immersed plane curves, known as the curve diffusion flow. It is known that under this flow there exist both initially immersed curves that develop at least one singularity in finite time and initially embedded curves that self-intersect in finite time. We prove that under the flow closed curves with initial data close to a round circle in the sense of normalised L ² oscillation of curvature exist for all time and converge exponentially fast to a round circle. This implies that for a sufficiently large ‘waiting time’, the evolving curves are strictly convex. We provide an optimal estimate for this waiting time, which gives a quantified feeling for the magnitude to which the maximum principle fails. We are also able to control the maximum of the multiplicity of the curve along the evolution. A corollary of this estimate is that initially embedded curves satisfying the hypotheses of the global existence theorem remain embedded. Finally, as an application we obtain a rigidity statement for closed planar curves with winding number one.     

Regularity of subelliptic Monge-Ampère equations

In dimension n?3, for k≈|x|^2m that can be written as a sum of squares of smooth functions, we prove that a C² convex solution u to a subelliptic Monge-Ampère equation detD²u=k(x,u,Du) is itself smooth if the elementary (n−1)st symmetric curvature kn−1 of u is positive (the case m?2 uses an additional nondegeneracy condition on the sum of squares). Our proof uses the partial Legendre transform, Calabi's identity for ∑uijσij where σ is the square of the third order derivatives of u, the Campanato method Xu and Zuily use to obtain regularity for systems of sums of squares of Hörmander vector fields, and our earlier work using Guan's subelliptic methods.     

Almost everywhere divergent subsequences of Fourier sums of functions from φ(L) ∩ H 1 ω

For a gap sequence of natural numbers {n _k } _k=1 ^∞ , for a nondecreasing function φ: [0,+∞) → [0,+∞) such that φ(u) = o(u ln ln u) as u → ∞, and a modulus of continuity satisfying the condition (ln k)^?1 = O(ω(n _k ^?1 )), we present an example of a function F ∈ φ(L) ∩ H ₁ ^ω with an almost everywhere divergent subsequence {S _{n k} (F, x)} of the sequence of partial sums of the trigonometric Fourier series of the function F.     

Property (X) for Second-Order Nonlinear Differential Equations of Generalized Euler Type

In this paper the generalized nonlinear Euler differential equation t²k(tu′)u″ + t(f(u)+ k(tu′))u′ + g(u) = 0 is considered. Here the functions f(u), g(u) and k(u) satisfy smoothness conditions which guarantee the uniqueness of solutions of initial value problems, however, no conditions of sub(super) linearity are assumed. We present some necessary and sufficient conditions and some tests for the equivalent planar system to have or fail to have property (X⁺), which is very important for the existence of periodic solutions and oscillation theory.     

Evolution of highly symmetric curves under the shrinking curvature flow

Under the shrinking curvature flow with inner normal velocity V = k^α(α > 1), it is shown that highly symmetric, locally convex initial curves evolve into a point asymptotically like an multi‐circles. The proof relies on a crucial use of Bonnensen inequality for highly symmetric, locally convex curves. Copyright © 2016 John Wiley & Sons, Ltd.     

An extremal case of the equation of prescribed Weingarten curvature

In this paper we investigate the existence and regularity of solutions to a Dirichlet problem for a Hessian quotient equation on the sphere. The equation in question arises as the determining equation for the support function of a convex surface which is required to meet a given enclosing cylinder tangentially and whose k-th Weingarten curvature is a given function. This is a generalization of a Gaussian curvature problem treated in [13]. Essentially given ${\Omega \subset \mathbb{R}^n}$ we seek a convex function u such that graph(u) has a prescribed k-th curvature ψ and |Du(x)| → ∞ as x → ?Ω. Under certain regularity assumptions on ψ and Ω we are able to demonstrate the existence of a solution whose graph is C ^3,α provided that ${\psi^{-\frac{1}{k}} = \psi^{-\frac{1}{k}}(x, \nu)}$ is convex in x and a certain compatibility condition between ψ| _?Ω and Ω is satisfied.     

On Hanner type inequalities with a weight for Banach spaces

We shall present several Hanner type inequalities with a weight constant and characterize 2-uniformly smooth and 2-uniformly convex Banach spaces with these inequalities. p-Uniformly smooth and q-uniformly convex Banach spaces will be also characterized with another Hanner type inequalities with a weight in the other side term. The best value of the weight in these inequalities will be determined for Lp spaces. Also we shall present a duality theorem between these inequalities in a generalized form.     

A sharp threshold for minimum bounded-depth and bounded-diameter spanning trees and Steiner trees in random networks

In the complete graph on n vertices, when each edge has a weight which is an exponential random variable, Frieze proved that the minimum spanning tree has weight tending to ζ(3) = 1/1³ + 1/2³ + 1/3³ +… as n → ∞. We consider spanning trees constrained to have depth bounded by k from a specified root. We prove that if k ≥ log₂ logn+ω(1), where ω(1) is any function going to ∞ with n, then the minimum bounded-depth spanning tree still has weight tending to ζ(3) as n → ∞, and that if k < log₂ logn, then the weight is doubly-exponentially large in log₂ logn ? k. It is NP-hard to find the minimum bounded-depth spanning tree, but when k≤log₂ logn?ω(1), a simple greedy algorithm is asymptotically optimal, and when k ≥ log₂ logn+ω(1), an algorithm which makes small changes to the minimum (unbounded depth) spanning tree is asymptotically optimal. We prove similar results for minimum bounded-depth Steiner trees, where the tree must connect a specified set of m vertices, and may or may not include other vertices. In particular, when m=const×n, if k≥log₂ logn+ω(1), the minimum bounded-depth Steiner tree on the complete graph has asymptotically the same weight as the minimum Steiner tree, and if 1 ≤ k ≤ log₂ logn?ω(1), the weight tends to $(1 - 2^{ - k} )\sqrt {8m/n} \left[ {\sqrt {2mn} /2^k } \right]^{1/(2^k - 1)}$ in both expectation and probability. The same results hold for minimum bounded-diameter Steiner trees when the diameter bound is 2k; when the diameter bound is increased from 2k to 2k+1, the minimum Steiner tree weight is reduced by a factor of $2^{1/(2^k - 1)}$ .     

Existence of positive periodic solutions of nonlinear first-order delayed differential equations

We consider the existence of positive ω-periodic solutions for the equation

u^′(t)=a(t)g(u(t))u(t)−λb(t)f(u(t−τ(t))),     

Keller-Osserman conditions for diffusion-type operators on Riemannian manifolds

In this paper we obtain essentially sharp generalized Keller-Osserman conditions for wide classes of differential inequalities of the form Lu?b(x)f(u)?(|∇u|) and Lu?b(x)f(u)?(|∇u|)−g(u)h(|∇u|) on weighted Riemannian manifolds, where L is a non-linear diffusion-type operator. Prototypical examples of these operators are the p-Laplacian and the mean curvature operator. The geometry of the underlying manifold is reflected, via bounds for the modified Bakry-Emery Ricci curvature, by growth conditions for the functions b and ?. A weak maximum principle which extends and improves previous results valid for the φ-Laplacian is also obtained. Geometric comparison results, valid even in the case of integral bounds for the modified Bakry-Emery Ricci tensor, are presented.     

Rate of convergence of the mean curvature flow

We study the flow M_t of a smooth, strictly convex hypersurface by its mean curvature in ?^{n + 1}. The surface remains smooth and convex, shrinking monotonically until it disappears at a critical time T and point x^* (which is due to Huisken). This is equivalent to saying that the corresponding rescaled mean curvature flow converges to a sphere Sⁿ of radius √n. In this paper we will study the rate of exponential convergence of a rescaled flow. We will present here a method that tells us that the rate of the exponential decay is at least 2/n. We can define the “arrival time” u of a smooth, strictly convex, n‐dimensional hypersurface as it moves with normal velocity equal to its mean curvature via u(x) = t if x ∈ M_t for x ∈ Int(M₀). Huisken proved that, for n ≥ 2, u(x) is C² near x^*. The case n = 1 has been treated by Kohn and Serfaty [11]; they proved C³‐regularity of u. As a consequence of the obtained rate of convergence of the mean curvature flow, we prove that u is not necessarily C³ near x^* for n ≥ 2. We also show that the obtained rate of convergence 2/n, which arises from linearizing a mean curvature flow, is the optimal one, at least for n ≥ 2. © 2007 Wiley Periodicals, Inc.     

Preservation of a.c. spectrum for random decaying perturbations of square-summable high-order variation

We consider random self-adjoint Jacobi matrices of the form

(Jωu)(n)=an(ω)u(n+1)+bn(ω)u(n)+an−1(ω)u(n−1)     

k-eccentricity and absolute k-centrum of a probabilistic tree

The k-eccentricity evaluated at a point x of a graph G is the sum of the (weighted) distances from x to the k vertices farthest from it. The k-centrum is the set of vertices for which the k-eccentricity is a minimum. The concept of k-centrum includes, as a particular case, that of center and that of centroid (or median) of a graph. The absolute k-centrum is the set of points (not necessarily vertices) for which the k-eccentricity is a minimum. In this paper it will be proven that, for a weighted tree, both deterministic and probabilistic, the k-eccentricity is a convex function and that the absolute k-centrum is a connected set and is contained in an elementary path. Hints will be given for the construction of an algorithm to find the k-centrum and the absolute k-centrum.     

The fairing and G1 continuity of quartic C‐Bézier curves

Quartic C‐Bézier curves possess similar properties with the traditional Bézier curves including terminal property, convex hull property, affine invariance, and approaching the shape of their control polygons as the shape parameter α decreases. In this paper, by adjusting the shape parameter α on the basis of the utilization of the least square approximation and nonlinear functional minimization together with fairing of a quartic C‐Bézier curve with G¹ continuity of quartic C‐Bézier curve segments, we develop a fairing and G¹ continuity algorithm for any given stitching coefficients λ_k(k = 1,2,…,n ? 1). The shape parameters α_i(i=1, 2, …, n) can be adjusted by the value of control points. The curvature of the resulting quartic C‐Bézier curve segments after fairing is more uniform than before. Moreover, six examples are provided in the paper to demonstrate the efficacy of the algorithm and illustrate how to apply this algorithm to the computer‐aided design/computer‐aided manufacturing modeling systems. Copyright © 2015 John Wiley & Sons, Ltd.     

The condition on the stability of stationary lines in a curvature flow in the whole plane

The long time behavior of a curve in the whole plane moving by a curvature flow is studied. Studying the Cauchy problem, we deal with moving curves represented by entire graphs on the x-axis. Here the initial curves are given by bounded functions on the x-axis. It is proved that the solution converges uniformly to the solution of the Cauchy problem of the heat equation with the same initial value. The difference is of order O(t−1/2) as time goes to infinity. The proof is based on the decay estimates for the derivatives of the solution. By virtue of the stability results for the heat equation, our result gives the sufficient and necessary condition on the stability of constant solutions that represent stationary lines of the curvature flow in the whole plane.     

Extinction and positivity of the solutions of the heat equations with absorption on networks

In this paper, we propose a discrete version of the following semilinear heat equation with absorption ut=Δu−uq with q>1, which is said to be the ω-heat equation with absorption on a network. Using the discrete Laplacian operator Δω on a weighted graph, we define the ω-heat equations with absorption on networks and give their physical interpretations. The main concern is to investigate the large time behaviors of nontrivial solutions of the equations whose initial data are nonnegative and the boundary data vanish. It is proved that the asymptotic behaviors of the solutions u(x,t) as t tends to +∞ strongly depend on the sign of q−1.