Jann-Long Chern

Smoothness and Periodicity of Some Matrix Decompositions

In this work we consider smooth orthonormal factorizations of smooth matrix-valued functions of constant rank. In particular, we look at Schur, singular value, and related decompositions. Furthermore, we consider the case in which the functions are periodic and study periodicity of the factors. We allow for eigenvalues and singular values to coalesce.

The symmetry of least-energy solutions for semilinear elliptic equations

Abstract In this paper we will apply the method of rotating planes (MRP) to investigate the radial and axial symmetry of the least-energy solutions for semilinear elliptic equations on the Dirichlet and Neumann problems, respectively. MRP is a variant of the famous method of moving planes. One of our main results is to consider the least-energy solutions of the following equation: (∗) Δ u+K(x)u p =0, x∈B 1 , u>0 in B 1 , u| ∂B 1 =0, where 1 n+2 n−2 and B1 is the unit ball of R n with n⩾3. Here K(x)=K(|x|) is not assumed to be decreasing in |x|. In this paper, we prove that any least-energy solution of (∗) is axially symmetric with respect to some direction. Furthermore, when p is close to n+2 n−2 , under some reasonable condition of K, radial symmetry is shown for least-energy solutions. This is the example of the general phenomenon of the symmetry induced by point-condensation. A fine estimate for least-energy solution is required for the proof of symmetry of solutions. This estimate generalizes the result of Han (Ann. Inst. H. Poincare Anal. Nonlineaire 8 (1991) 159) to the case when K(x) is nonconstant. In contrast to previous works for this kinds of estimates, we only assume that K(x) is continuous.

Existence of positive entire solutions of some semilinear elliptic equations

in R”, n 2 3, where K is a suitable function on R”, m is a real constant not equal to one, and A = x1=, d2/axf. We are concerned with the problem of finding positive solutions u of (1 .l ) on 08”. The equation (1.1) arises both in geometry and in physics. When m = (n + 2)/(n 2), (1.1) is now known as the conformal scalar curvature equation in R”. When K = 1, (1.1) is known as the Lane-Emden equation in astrophysics or sometimes the Emden-Fowler equation. In this case, u corresponds to the density of a single star, and positive radial solutions of (1.1) in balls with zero Dirichlet boundary data are of particular interest. When n = 3, m > 1, and K(x) = l/(1 + lx12), (1.1) was proposed by Matukuma [M] in 1930 as a mathematical model to describe the dynamics of a globular cluster of stars. In this context, u represents the gravitational potential (therefore u>O), p = (41~)-l( 1 + [xl’)-’ zP represents the density and SW3 p dx represents the total mass of the cluster of stars. Since the globular cluster has a radial symmetry, positive radial entire solutions (i.e., solutions with u = u( 1x1) > 0

On the Classification of Standing Wave Solutions for the Schrödinger Equation

In this article, we consider the following semilinear elliptic equation where n ≥ 3, p > 1 and λ > 0. We provide the existence and uniqueness of the singular radial solution of the above equation for specific ranges of n and p. In addition, we also clarify the entire structure of radial solutions for various types according to their behaviors at the origin and infinity.

Existence, uniqueness and stability of positive solutions to sublinear elliptic systems

Reaction–diffusion systems are used to model many chemical and biological phenomena in the natural world [25, 26], and systems of coupled partial differential equations are also used in other physical models such as nonlinear Schrodinger systems in multi-component Bose–Einstein condensates and nonlinear optics [19, 23]. The steady-state solutions or standing-wave solutions of such systems of nonlinear partial differential equations satisfy a nonlinear elliptic system with more than one equation. Much effort has been devoted to the existence of solutions of such systems (see, for example, [4, 8, 10, 11, 13, 15–17, 24, 30, 33, 34]), but it is usually difficult to determine whether or not the solution is unique. We consider the positive solutions of a semilinear elliptic system of the form

Uniqueness of finite total curvatures and the structure of radial solutions for nonlinear elliptic equations

In this article, we are concerned with the semilinear elliptic equation where n > 2, p > 1, and K(lxl) > 0 in R n . The correspondence between the initial values of regularly positive radial solutions of the above equation and the associated finite total curvatures will be derived. In addition, we also conduct the zeros of radial solutions in terms of the initial data under specific conditions on K and p. Furthermore, based on the Pohozaev identity and openness for the regions of initial data corresponding to certain types of solutions, we obtain the whole structure of radial solutions depending on various situations.

The solution structure of the O(3) sigma model in a Maxwell-Chern-Simons theory

In this paper, a system of semilinear elliptic equations arising from a relativistic self-dual Maxwell-Chern-Simons O(3) sigma model is considered. We reveal the uniqueness aspect of the topological solutions for the model. The uniqueness result is associated with a clear solution structure of the equations of the radially symmetric case. We locate each solution set denoted by a planar diagram.

A survey of solutions in a gravitational Born-Infeld theory

An elliptic equation that arises from a cosmic string model with the action of the Born-Infeld nonlinear electromagnetism, is considered. We classify and establish the uniqueness of radially symmetric solutions.

Topological multivortex solutions for the Chern–Simons system with two Higgs particles

ABSTRACT In this paper, we prove the uniqueness of topological multivortex solutions to the self-dual abelian Chern–Simons model if either the Chern–Simons coupling parameter is sufficiently small or sufficiently large. In addition, we also establish the sharp region of the flux for nontopological solutions with a single vortex point.

Singular Solutions of the Scalar Field Equation with a Critical Exponent

We consider radially symmetric singular solutions of the scalar field equation with the Sobolev critical exponent. It is shown that there exists a unique special singular solution, and other infinitely many singular solutions are oscillatory around the special singular solution.