Selected Papers of Weiyue Ding

The following sections are included:

• Preface
• Contents
• Biography

The following sections are included:

• INTRODUCTION
• A GENERAL PRINCIPLE
• EXISTENCE THEOREM FOR LIENARD SYSTEM
• REFERENCES

A generalized form of the Poincaré-Birkhoff theorem is proved. The generalization is useful for the further applications of this famous fixed point theorem.

CONSIDER

The following sections are included:

• Introduction
• Preliminaries
• An existence result
• Some a priori estimates
• The radial case
• REFERENCES

The following sections are included:

• INTRODUCTION
• PROOF OF THEOREM I
• PROOFS OF THEOREM II AND III
• BANACH SPACE EXTENSION AND AN APPLICATION
• REFERENCES

Consider the Duffing’s equation

Under suitable hypotheses we obtain various theorems concerning the existence of positive solutions of the equation

For n ≧ 3, the equation Δu+ |u|^{4/(n − 2)}u = 0 on ℝⁿ has infinitely many distinct solutions with finite energy and which change sign.

The following sections are included:

• Introduction
• Preliminaries on Lusternik-Schnirelmann Category
• Lsternik-Schnirelmann Theory via Perturbation Methods
• Applications to Harmonic Maps
• REFERENCES

A theorem for the existence of solutions of the nonlinear elliptic equation –Δu + 2 = R(x)e^u, x ∊ S ², is proved by using a “mass center” analysis technique and by applying a continuous “flow” in H¹ (S²) controlled by ∇R.

Let R(x) be a smooth function on the 2-sphere S². A question m differential geometry may be raised: Can R(x) be the scalar curvature of a metric g on S² which is pointwise conformal to the standard metric g₀ (i. e. g = e^ug₀)? This problem can be reduced to solving the following elliptic equation

Necessary and sufficient conditions for the existence of symmetric harmonic maps between spheres are established.

Recently, there has been interesting progress on the problem of existence of Kähler-Einstein metrics on compact Kähler manifolds with positive first Chern class (cf. [Si, Ti, T-Y]). The approach proposed by Tian is of particular interest In [Ti], he defined an invariant α(M) for compact Kähler manifolds with c₁(M)>0, and proved that, if $\alpha \left(M\right)>\frac{n}{n+1},$ where n = dimM, then there exists a Kählern Einstein metric on M. The definition of α(M) is as follows…

We show that there are bounded contractible domains in Rⁿ, n ≥ 3, on which the Dirichlet problem for the equation Δu+u^(u+2)/(u-2) = 0 admit positive solutions. This result, combining with the well-known nonexistence result of Pohozaev, implies that geometry of the domains plays a crucial role in the solvability of the problem.

In this paper it is proved that the solution to the evolution problem for harmonic maps blows up in finite time, if the initial map belongs to some nontrivial homotopy class and the initial energy is sufficiently small.

Let (M, g) and (N, h) be two Riemannian manifolds. Consider the heat flow for harmonic maps from (M, g) into (N, h). We prove the following result: Suppose dim M = 3 and $\mathcal{F}$ is a nontrivial homotopy class in C(M, N). Then there exists a constant ∊ > 0 such that if u₀ ∊ $\mathcal{F}$ and E(u₀) < ∊, the solution of the heat flow with initial value u₀ must blow up in finite time. We also present a sufficient condition which ensures that any global solutions subconverge to harmonic maps as t → ∞

The following sections are included:

• Introduction
• Preliminaries on Estimates for Harmonic Maps
• The Main Existence Theorems
• Applications
• References

The following sections are included:

• Introduction
• Solutions with Symmetric Initial Value
• Proof of Theorem 1.1
• References

It is investigated that in higher dimensions the heat flow of harmonic maps between two compact Riemannian manifolds blows up in a finite time if the initial map is in some nontrivial homotopic class with small energy.

In this talk we report some recent results on the heat flow approach to the study of harmonic maps…

Let X be a compact Kähler manifold with positive first Chern class, and let ω be a fixed Kähler metric on X. In this note, we study a functional F_ω(·) on the space of all Kähler metrics which are cohomologous to ω. Such a functional was first studied by J. Moser for X = ℂ P¹ = S² with the standard metric, and its boundedness from below is known as the Moser-Trudinger inequality. We prove that the generalized Moser-Trudinger inequality holds on X, provided X admits Kähler-Einstein metrics. We also prove that, if F_ω is bounded from below, then F_ω(φ_t) converge to the absolute minimum of F_ω as t→ 1, where φ_t is the solutions of the so-called Aubin’s flow. Finally, relations between F_ω and the K-energy functional introduced by Mabuchi are discussed.

This note is concerned with the singular behaviour of solutions to the heat equation of harmonic maps between Riemannian manifolds. Eells and Sampson [5] showed that for C¹ initial values the solutions exist locally in time and asked whether they exist for all time or they may blow up in finite time. Recently, Coron-Ghidaglia [3], Ding [4] and Chen-Ding [2] constructed many examples of finite-time blow-up of the solutions in the case where the domain manifold has dimension greater than two…

We generalize the well-known Eells-Sampson’s theorem on the global existence and convergence for the heat flow of harmonic maps. The assumption that the curvature of the target manifold N be nonpositive is replaced by the weaker one requiring that the universal cover Ñ admit a strictly convex function with quadratic growth.

The following sections are included:

• Introduction
• The lower boundedness of the K-energy
• The proof of the main theorems
• Examples
• Some remarks
• References

The following sections are included:

• Introduction
• Analysis of the O.D.E.
• Variational Analysis
• References

By introducing the “relative energy”, we develop a new method for finding harmonic maps from noncompact complete Riemannian manifolds with prescribed asympototic behaviour at infinity. This method is an extension of the well known direct method of energy-minimization for compact domains. As an application of our method, we show that the Dirichlet problem at infinity with Hölder continuous boundary data for harmonic maps from a Cartan-Hadarmard manifold with bounded negative curvature into a compact manifold, has a locally minimizing solution which is smooth near infinity.

The problem of existence of conformal metrics with Gaussian curvature equal to a given function K on a compact Riemannian 2-manifold M of negative Euler characteristic is studied. Let K₀ be any nonconstant function on M with max K₀ = 0, and let K_λ = K₀ + λ. It is proved that there exists a λ^* > 0 such that the problem has a solution for K = K_λ iff λ ∊ (-∞, λ*]. Moreover, if λ ∊ (0, λ*), then the problem has at least 2 solutions.

The following sections are included:

• Introduction
• Proof of Theorem 2
• Two Remarks
• References

We give a simple proof of the well-known Hamilton’s result [1] on the heat flows and harmonic maps from manifolds with boundary using the approach of Ding-Lin [2].

Let M be a compact Riemann surface, h(x) a positive smooth function on M. In this paper, we consider the functional

Extending work of Caffarelli-Yang and Tarantello, we present a variational existence proof for two-vortex solutions of the periodic Chern-Simons Higgs model and analyze the asymptotic behavior of these solutions as the parameter coupling the gauge field with the scalar field tends to 0.

The definition of Schrödinger flow is proposed. It is indicated that the flow of ferromagnetic chain is actually Schrödinger flow of maps into S², and that there exists a unique local smooth solution for the initial value problem of one-dimensional Schrödinger flow of maps into Kähler manifolds. In case the targets are Kähler manifolds with constant curvature, it is proved that one-dimensional Schrödinger flow admits a unique global smooth solution.

We consider a Ginzburg-Landau type functional on S² with a 6^th order potential and the corresponding selfduality equations. We study the limiting behavior in the two vortex case when a coupling parameter tends to zero. This two vortex case is a limiting case for the Moser inequality, and we correspondingly detect a rich and varied asymptotic behavior depending on the position of the vortices. We exploit analogies with the Nirenberg problem for the prescribed Gauss curvature equation on S².

We present a simple proof for the uniformization theorem on 2-sphere by methods of elliptic partial differential equations.

Let Ω be an annulus. We prove that the mean field equation

The abelian Chern–Simons–Higgs model of Hong-Kim-Pac and Jackiw–Weinberg leads to a Ginzburg–Landau type functional with a 6^th order potential on a compact Riemann surface. We derive the existence of two solutions with different asymptotic behavior as the coupling parameter tends to 0, for any number of prescribed vortices. We also introduce a Seiberg–Witten type functional with a 6^th order potential and again show the existence of two asymptotically different solutions on a compact Kähler surface. The analysis is based on maximum principle arguments and applies to a general class of scalar equations.

In this paper we show that there exists a unique local smooth solution for the Cauchy problem of the Schrödinger flow for maps from a compact Riemannian manifold into a complete Kähler manifold, or from a Euclidean space R^m into a compact Kähler manifold. As a consequence, we prove that Heisenberg spin system is locally well-posed in the appropriate Sobolev spaces.

We present some recent results on the existence of solutions of the Schrödinger flows, and pose some problems for further research.

The following sections are included:

• Introduction and the main result
• Blow-up analysis
• Comparison
• Proof of the proposition
• References

Let M and N be two compact Riemannian manifolds. Let u^k be a sequence of stationary harmonic maps from M to N with bounded energies. We may assume that it converges weakly to a weakly harmonic map u in H^{1, 2}(M, N) as k → ∞. In this paper, we construct an example to show that the limit map u may not be stationary.

In this note, we prove that the Schrödinger flow of maps from a closed Riemann surface into a compact irreducible Hermitian symmetic space admits a global weak solution. Also, we show the existence of weak solutions to the initial value problem of Heisenberg model with Lie algebra values, which is closely related to the Schrödinger flow on compact Hermitian symmetric spaces.

The following sections are included:

• Introduction
• The reduced variational problem for N = S²
• An Existence Theorem
• The case of M = S²
• Final Remarks
• References

The following sections are included:

• Introduction
• Basic concepts and properties
• Long time existence
• Behavior near infinity
• References

In this paper we will mainly propose some problems for a class of degenerate elliptic equations, either linear or nonlinear. We will study some special cases of these problems and reveal some phenomena which may not have been noticed previously.

Our problems originated from the self-similar solutions of the heat flow of harmonic maps. We will prove that the self-similar solutions or the so-called quasi-harmonic spheres are discontinuous at infinity for the equivariant case. In other words, the equivariant quasi-harmonic spheres are not continuous images of topological spheres.

We resolve the singularities of the minimal Hopf cones by families of regular minimal graphs.

It is proved that given a conformal metric e^u0g₀, with e^u0 ∈ L^∞, on a 2-dim closed Riemannian manfold (M, g₀), there exists a unique smooth solution u(t) of the Ricci flow such that u(t) → u₀ in L² as t → 0.

In this note, we showed the existence of equivariant self-similar solutions with finite local energy for the Schrödinger flow from ℂⁿ into ℂPⁿ (n≥2).

Inspired by the construction of blow-up solutions of the heat flow of harmonic maps from D² into S² via maximum principle (Chang et al., J. Diff. Geom., 36, 1992, pp. 507-515.) we provide examples of nonexistence of smooth axially symmetric harmonic maps from B³ into S² with smooth boundary maps of degree zero.