Infinitely many non-radial solutions for fractional Nirenberg problem

In this paper, the prescribed \(\sigma \)-curvature problem

$$\begin{aligned} P_{\sigma }^{g_0} u={\tilde{K}}(x)u^{\frac{N+2\sigma }{N-2\sigma }}, x\in {\mathbb {S}}^N,u>0 \end{aligned}$$

is considered. When \({\tilde{K}}(x)\) is some axis symmetric function on \({\mathbb {S}}^N\), by using singular perturbation method, it is proved that this problem possesses infinitely many non-radial solutions for \(0<\sigma \le 1\) and \(N> 2\sigma +2\).

     

On a problem of Nirenberg concerning expanding maps

Let X be a Banach space and T:X → X a continuous map, which is expanding (i.e., ∥Tu ? Tv∥ ? ∥u ? v∥ for all u, v?X) and such that T(X) has a nonempty interior. Does this guarantee that T is onto? We give a counterexample in the case of X=L¹(N).     

The Brezis–Nirenberg problem for fractional elliptic operators

Let be a uniformly elliptic operator in divergence form in a bounded open subset Ω of . We study the effect of the operator on the existence and nonexistence of positive solutions of the nonlocal Brezis–Nirenberg problem where denotes the fractional power of with zero Dirichlet boundary values on , , and λ is a real parameter. By assuming for all and near some point , we prove existence theorems for any , where denotes the first Dirichlet eigenvalue of . Our existence result holds true for and in the interior case () and for and in the boundary case (). Nonexistence for star‐shaped domains is obtained for any .     

Curvature estimates for the Ricci flow II

In this paper we present several curvature estimates and convergence results for solutions of the Ricci flow, including the volume normalized Ricci flow and the normalized Kähler-Ricci flow. The curvature estimates depend on smallness of certain local space-time integrals of the norm of the Riemann curvature tensor, while the convergence results require finiteness of space-time integrals of this norm. These results also serve as characterization of blow-up singularities.     

Curvature estimates for the Ricci flow I

In this paper we present several curvature estimates for solutions of the Ricci flow and the modified Ricci flow (including the volume normalized Ricci flow and the normalized Kähler-Ricci flow), which depend on the smallness of certain local \(L^{\frac{n}{2}}\) integrals of the norm of the Riemann curvature tensor |Rm|, where n denotes the dimension of themanifold. These local integrals are scaling invariant and very natural.     

A flow method to the Orlicz-Aleksandrov problem

In this paper, we obtain an existence result of smooth solutions to the Orlicz-Aleksandrov problem from the perspective of geometric flow. Furthermore, a special uniqueness result of solutions to this problem shall be discussed.     

Curvature integrals under the Ricci flow on surfaces

In this paper, we consider the behavior of the total absolute and the total curvature under the Ricci flow on complete surfaces with bounded curvature. It is shown that they are monotone non-increasing and constant in time, respectively, if they exist and are finite at the initial time. As a related result, we prove that the asymptotic volume ratio is constant under the Ricci flow with non-negative Ricci curvature, at the end of the paper.      

Correction to: Using the minimum maximum flow degree to approximate the flow coloring problem

Annals of Operations Research -     

Solving the resource constrained deadline scheduling problem via reduction to the network flow problem

In this paper the problem of the equivalence of scheduling tasks on processors with the presence of deadlines and additional resources and the network flow problem is studied. This equivalence permits it to be proven that the problem of finding a maximal flow in a binary network with multipliers equal to 1 or 2 is NP-complete.     

A contribution to the stochastic flow shop scheduling problem

This paper deals with performance evaluation and scheduling problems in m machine stochastic flow shop with unlimited buffers. The processing time of each job on each machine is a random variable exponentially distributed with a known rate. We consider permutation flow shop. The objective is to find a job schedule which minimizes the expected makespan. A classification of works about stochastic flow shop with random processing times is first given. In order to solve the performance evaluation problem, we propose a recursive algorithm based on a Markov chain to compute the expected makespan and a discrete event simulation model to evaluate the expected makespan. The recursive algorithm is a generalization of a method proposed in the literature for the two machine flow shop problem to the m machine flow shop problem with unlimited buffers. In deterministic context, heuristics (like CDS [Management Science 16 (10) (1970) B630] and Rapid Access [Management Science 23 (11) (1977) 1174]) and metaheuristics (like simulated annealing) provide good results. We propose to adapt and to test this kind of methods for the stochastic scheduling problem. Combinations between heuristics or metaheuristics and the performance evaluation models are proposed. One of the objectives of this paper is to compare the methods together. Our methods are tested on problems from the OR-Library and give good results: for the two machine problems, we obtain the optimal solution and for the m machine problems, the methods are mutually validated.     

A flow approach to the planar Lp$L_p$ Minkowski problem

This paper will establish the asymptotic behavior of an anisotropic area-preserving flow which shows the existence of smooth solutions of the planar $L_p$ Minkowski problem for $p\ne 0$.     

Another approach to the heat flow for Plateau problem

For the minimal surfaces in Rⁿ with Plateau boundary condition and establish the global existence and uniqueness of the flow as well as the continuous dependence of the initial datum.     

An algorithm to solve the proportional network flow problem

The proportional network flow problem is a generalization of the equal flow problem on a generalized network in which the flow on arcs in given sets must all be proportional. This problem appears in several natural contexts, including processing networks and manufacturing networks. This paper describes a transformation on the underlying network that reduces the problem to the equal flow problem; this transformation is used to show that algorithms that solve the equal flow problem can be directly applied to the proportional network flow problem as well, with no increase in asymptotic running time. Additionally, computational results are presented for the proportional network flow problem demonstrating equivalent performance to the same algorithm for the equal flow problem.     

Curvature flow of complete hypersurfaces in hyperbolic space

In this paper we continue our study of finding the curvature flow of complete hypersurfaces in hyperbolic space with a prescribed asymptotic boundary at infinity. Our main results are proved by deriving a priori global gradient estimates and C ² estimates.     

On the equivalence of the maximum balanced flow problem and the weighted minimax flow problem

We point out the equivalence of the maximum balanced flow problem of Minoux and the weighted minimax flow problem of Ichimori, Ishü and Nishida. Some generalizations of the two problems are also suggested.     

Multicommodity flow problem

The problem of nonmixing multicommodity flow is investigated. The theorem on the number of nonmixing flows of different commodities passing simultaneously through an oriented network is proved.Translated from Dinamicheskie Sistemy, No. 5, pp. 95–97, 1986.     

Curvature flow with driving force on fixed boundary points

In this paper, we consider the curvature flow with driving force on fixed boundary points in the plane. We give a general local existence and uniqueness result of this problem with \(C^2\) initial curve. For a special family of initial curves, we classify the solutions into three categories. Moreover, in each category, the asymptotic behavior is given.