Gamma limit of the nonself-dual Chern-Simons-Higgs energy

We consider the Gamma limit of the Abelian Chern-Simons-Higgs energy

     

A system of elliptic equations arising in Chern-Simons field theory

We prove the existence of topological vortices in a relativistic self-dual Abelian Chern-Simons theory with two Higgs particles and two gauge fields through a study of a coupled system of two nonlinear elliptic equations over R². We present two approaches to prove existence of solutions on bounded domains: via minimization of an indefinite functional and via a fixed point argument. We then show that we may pass to the full R² limit from the bounded-domain solutions to obtain a topological solution in R².     

Local existence of solutions for the Neumann problem of the nonlinear elastodynamic system outside a domain

In this paper, we study the mixed initial-boundary value problem of Neumann type for the nonlinear elastic wave equation outside a domain. The local existence of solutions to this problem is proved by iteration. To get this result, we prove the existence of solutions for the second order linear hyperbolic system with variable coefficients (in Sobolev spaces) outside of a domain by using linear evolution operators and integro-differential equations.     

Infinitely many solutions for some nonlinear scalar system of two elliptic equations

In this paper, we consider the elliptic system of two equations in H¹(RN)×H¹(RN):

     

Local Hopf bifurcation and global periodic solutions in a delayed predator-prey system

We consider a delayed predator-prey system. We first consider the existence of local Hopf bifurcations, and then derive explicit formulas which enable us to determine the stability and the direction of periodic solutions bifurcating from Hopf bifurcations, using the normal form theory and center manifold argument. Special attention is paid to the global existence of periodic solutions bifurcating from Hopf bifurcations. By using a global Hopf bifurcation result due to Wu [Trans. Amer. Math. Soc. 350 (1998) 4799], we show that the local Hopf bifurcation implies the global Hopf bifurcation after the second critical value of delay. Finally, several numerical simulations supporting the theoretical analysis are also given.     

Global smooth solutions of 3-D null-form wave equations in exterior domains with Neumann boundary conditions

The paper is devoted to investigating long time behavior of smooth small data solutions to 3-D quasilinear wave equations outside of compact convex obstacles with Neumann boundary conditions. Concretely speaking, when the surface of a 3-D compact convex obstacle is smooth and the quasilinear wave equation fulfills the null condition, we prove that the smooth small data solution exists globally provided that the Neumann boundary condition on the exterior domain is given. One of the main ingredients in the current paper is the establishment of local energy decay estimates of the solution itself. As an application of the main result, the global stability to 3-D static compressible Chaplygin gases in exterior domain is shown under the initial irrotational perturbation with small amplitude.     

On the regularity of weak solutions of certain elliptic systems

For certain quasilinear elliptic systems with perturbations of natural growth we prove a Caccioppoli-inequality provided the perturbation satisfies an additional “angle condition.” As a consequence weak solutions of these systems have the Hölder-continuity properties established by the so called direct approach to regularity, cf.: [2].     

Local and global uniform convergence for elliptic problems on varying domains

The aim of the paper is to prove optimal results on local and global uniform convergence of solutions to elliptic equations with Dirichlet boundary conditions on varying domains. We assume that the limit domain be stable in the sense of Keldyš [Amer. Math. Soc. Transl. 51 (1966) 1-73]. We further assume that the approaching domains satisfy a necessary condition in the inside of the limit domain, and only require L²-convergence outside. As a consequence, uniform and L²-convergence are the same in the trivial case of homogenisation of a perforated domain. We are also able to deal with certain cracking domains.     

Explicit estimates for solutions of nonlinear radiation-type problems

We establish the existence of weak solutions of a nonlinear radiation-type boundary value problem for elliptic equation on divergence form with discontinuous leading coefficient. Quantitative estimates play a crucial role on the real applications. Our objective is the derivation of explicit expressions of the involved constants in the quantitative estimates, the so-called absolute or universal bounds. The dependence on the leading coefficient and on the size of the spatial domain is precise. This work shows that the expressions of those constants are not so elegant as we might expect.     

Differentiability of solutions to second-order elliptic equations via dynamical systems

For a second-order elliptic equation in divergence form we investigate conditions on the coefficients which imply that all solutions are Lipschitz continuous or differentiable at a given point. We assume the coefficients have modulus of continuity satisfying the square-Dini condition, and obtain additional conditions that examples show are sharp. Our results extend those of previous authors who assume the modulus of continuity satisfies the Dini condition. Our method involves the study of asymptotic properties of solutions to a dynamical system that is derived from the coefficients of the elliptic equation.     

Two-Dimensional Div-Curl Results: Application to the Lack of Nonlocal Effects in Homogenization

In this paper, we study the asymptotic behaviour of sequences of conduction problems and sequences of the associated diffusion energies. We prove that, contrary to the three-dimensional case, the boundedness of the conductivity sequence in L¹ combined with its equi-coerciveness prevents from the appearance of nonlocal effects in dimension two. More precisely, in the two-dimensional case we extend the Murat–Tartar H-convergence which holds for uniformly bounded and equi-coercive conductivity sequences, as well as the compactness result which holds for bounded and equi-integrable conductivity sequences in L¹. Our homogenization results are based on extensions of the classical div-curl lemma, which are also specific to the dimension two.     

Existence of solutions to a global eikonal equation

We derive a geometric necessary and sufficient condition for the existence of solutions to a global eikonal equation. We also study the existence of a minimal solution to this equation, and its relation with the well-known minimal time function.     

Uniform bound of Sobolev norms of solutions to 3D nonlinear wave equations with null condition

This article concerns the time growth of Sobolev norms of classical solutions to the 3D quasi-linear wave equations with the null condition. Given initial data in H^s×H^s−1$H^s\times H^{s-1}$ with compact supports, the global well-posedness theory has been established independently by Klainerman [13] and Christodoulou [3], respectively, for a relatively large integer s  . However, the highest order Sobolev energy, namely, the H^s$H^s$ energy of solutions may have a logarithmic growth in time. In this paper, we show that the H^s$H^s$ energy of solutions is also uniformly bounded for s?5$s?5$. The proof employs the generalized energy method of Klainerman, enhanced by weighted L²$L^2$ estimates and the ghost weight introduced by Alinhac.     

Infinitely many solutions for elliptic systems with critical exponents

One class of critical growth elliptic systems of two equations is considered on a bounded domain. By using fountain theorem and concentration estimates, we establish the existence of infinitely many solutions in higher values of dimension N?7.     

Local and global behaviour of steady-state solutions of the Sel'kov model

In this paper we discuss steady-state solutions of the systemof reaction-diffusion equations known as the Sel'kov model.This model has been the subject of much discussion; in particular,analytical and numerical results have been discussed by Lopez-Gomezet al. (1992, IMA J. Num. Anal. 12, 405–28). We show thata simple analysis of the bifurcation function associated withthe system can explain many of the numerical observations, suchas the formation and development of loops of nontrivial solutions,in a simpler and more complete manner than the analysis of Lopez-Gomezet al. This allows for a clearer understanding of the qualitativebehaviour of the set of nontrivial solutions and hence of thebifurcation diagram.