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Discrete & Continuous Dynamical Systems - A

2007 , Volume 19 , Issue 2

Special Issue on
Variational Problems and Applications

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Preface
Chjan C. Lim, Mitsuharu Ôtani and Junping Shi
2007, 19(2): i-iv doi: 10.3934/dcds.2007.19.2i +[Abstract](362) +[PDF](62.3KB)
Abstract:
Calculus of variation is one of the oldest yet still continually developing branches of analysis. Physical variational principles were discovered prior to the birth of modern calculus and associated with names such as Fermat. With the historical development of the differential and integral calculus of Newton and Leibniz, the principle of extreme action in mechanics was formulated with effort from Maupertuis, Euler and Lagrange in mid-eighteenth century, followed by the work of deriving the equations of motion from the variational principles by Jacobi and Hamilton nearly one hundred years later. Since then calculus of variation has become one of the main tools in analysis, with numerous applications in physical sciences and engineering.

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Dynamics of two interacting circular cylinders in perfect fluid
A. V. Borisov, I.S. Mamaev and S. M. Ramodanov
2007, 19(2): 235-253 doi: 10.3934/dcds.2007.19.235 +[Abstract](515) +[PDF](377.3KB)
Abstract:
In this paper we consider the system of two 2D rigid circular cylinders immersed in an unbounded volume of inviscid perfect fluid. The circulations around the cylinders are assumed to be equal in magnitude and opposite in sign. We also explore some special cases of this system assuming that the cylinders move along the line through their centers and the circulation around each cylinder is zero. A similar system of two interacting spheres was originally considered in the classical works of Carl and Vilhelm Bjerknes, H. Lamb and N. E. Joukowski.
    By making the radii of the cylinders infinitesimally small, we have obtained a new mechanical system which consists of two regular point vortices but with non-zero masses. The study of this system can be reduced to the study of the motion of a particle subject to potential and gyroscopic forces. A new integrable case is found. The Hamiltonian equations of motion for this system have been generalized to the case of an arbitrary number of mass vortices with arbitrary intensities. Some first integrals have been obtained. These equations expand upon the classical Kirchhoff equations of motion for $n$ point vortices.
Multi-peak standing waves for nonlinear Schrödinger equations with a general nonlinearity
Jaeyoung Byeon and Louis Jeanjean
2007, 19(2): 255-269 doi: 10.3934/dcds.2007.19.255 +[Abstract](658) +[PDF](213.1KB)
Abstract:
We consider singularly perturbed elliptic equations $\varepsilon^2\Delta u - V(x) u + f(u)=0, x\in R^N, N \ge 3.$ For small $\varepsilon > 0,$ we glue together localized bound state solutions concentrating at isolated components of positive local minimum of $V$ under conditions on $f$ we believe to be almost optimal.
Boundary blow-up solutions with interior layers and spikes in a bistable problem
Yihong Du, Zongming Guo and Feng Zhou
2007, 19(2): 271-298 doi: 10.3934/dcds.2007.19.271 +[Abstract](580) +[PDF](293.1KB)
Abstract:
We show that for small $\epsilon>0$, the boundary blow-up problem

$-\epsilon^2\Delta u= u (u-a(x))(1-u) \mbox{ in } \Omega, u|_{\partial\Omega}=\infty$

has solutions with sharp interior layers and spikes, apart from boundary layers. We also determine the location of these layers and spikes.

Vector p-Laplacian like operators, pseudo-eigenvalues, and bifurcation
Marta García-Huidobro, Raul Manásevich and J. R. Ward
2007, 19(2): 299-321 doi: 10.3934/dcds.2007.19.299 +[Abstract](526) +[PDF](250.2KB)
Abstract:
Boundary value problems for systems of ordinary differential equations are studied. These systems involve asymptotically homogeneous operators. Leray-Schauder indices are calculated for these operators and the concept of pseudo-eigenvalue is defined. The existence of nontrivial solutions is studied. Conditions for bifurcation, from either zero or infinity, at the pseudo-eigenvalues are given.
Existence of nontrivial solutions for some elliptic equations with supercritical nonlinearity in exterior domains
Satoshi Hashimoto and Mitsuharu Ôtani
2007, 19(2): 323-333 doi: 10.3934/dcds.2007.19.323 +[Abstract](446) +[PDF](160.5KB)
Abstract:
The existence of positive solutions is discussed for some nonlinear elliptic equations involving the nonlinear terms with the growth order of super-critical exponents in exterior domains of balls such as $ -\Delta u = u^\beta $ in $\Omega$, ($(N+2)/(N-2) < \beta $), $u = 0 $ on $\partial B$, with $\Omega = \mathbb{R}^N \setminus\overline\Omega_0$ where $\Omega_0$ is the open ball. To recover the compactness of the embedding $L^{\beta+1}(\Omega) \subset H^1_0(\Omega)$, we work in the class of radially symmetric functions and introduce a new transformation, which reduces our problems to some nonlinear elliptic equations in annuli but with coefficients which have some singularity on the boundary. The difficulty caused by the singularity on the boundary will be managed by the arguments developed in our previous work.
Elliptic-parabolic variational inequalities with time-dependent constraints
Masahiro Kubo and Noriaki Yamazaki
2007, 19(2): 335-359 doi: 10.3934/dcds.2007.19.335 +[Abstract](521) +[PDF](264.8KB)
Abstract:
We study variational inequalities for quasilinear elliptic-parabolic equations with time-dependent constraints. Introducing a general condition for the time-dependence of convex sets defining the constraints, we establish theorems concerning existence, uniqueness as well as an order property of solutions. Some applications of the general results are given.
Extremal free energy in a simple mean field theory for a coupled Barotropic fluid - rotating sphere system
Chjan C. Lim
2007, 19(2): 361-386 doi: 10.3934/dcds.2007.19.361 +[Abstract](440) +[PDF](275.1KB)
Abstract:
A family of spin-lattice models are derived as convergent finite dimensional approximations to the rest frame kinetic energy of a barotropic fluid coupled to a massive rotating sphere. A simple mean field theory for this statistical equilibrium model is formulated and solved, providing precise conditions on the planetary spin and relative enstrophy in order for phase transitions to occur at positive and negative critical temperatures, $T_{+}$ and $T_{-}.$ When the planetary spin is relatively small, there is a single phase transition at $T_{-}<0,$ from a preferred mixed vorticity state $v=m$ for all positive temperatures and $T < T_{-}$ to an ordered pro-rotating (west to east) flow state $v=n_u$ for $T_{-} < T <0.$ When the planetary spin is relatively large, there is an additional phase transition at $T_{+}>0$ from a preferred mixed state $v=m$ above $T_{+}$ to an ordered counter-rotating flow state $v=n_d$ for $T < T_{+}.$
An expository survey on the recent development of mean field equations
Chang-Shou Lin
2007, 19(2): 387-410 doi: 10.3934/dcds.2007.19.387 +[Abstract](525) +[PDF](287.7KB)
Abstract:
We consider mean field equations: $ \Delta u+\rho(\frac{he^u}{\int_Mhe^u}-1)=0\, on M, $ where $M$ is a compact Riemann surface with area 1, $h$ is a positive continuous function and $\rho$ is a constant, or

$\Delta u+\rho\frac{he^u}{\int_\Omega he^u}=0$ in $\Omega, $
$ u=0 on \partial\Omega, $

where $\Omega$ is a bounded $\mathcal{C}^1$ domain of $\mathbb{R}^2$. In this paper, we give a short survey on the uniqueness problem, find sharper estimates of bubbling solutions and count the topological degree of solutions.

N-vortex equilibrium theory
P.K. Newton
2007, 19(2): 411-418 doi: 10.3934/dcds.2007.19.411 +[Abstract](490) +[PDF](133.9KB)
Abstract:
The problem of finding and classifying all relative equilibrium configurations of $N$-point vortices in the plane is first described as a classical variational principle and then formulated as a problem in linear algebra. Given a configuration of $N$ points in the plane, one must understand the structure of the $N(N-1)/2 \times N$ configuration matrix $A$ obtained by requiring that all interparticle distances remain fixed in time. If the determinant of the square, symmetric $N \times N$ covariance matrix $A^T A$ is zero, there is a non-trivial nullspace of $A$ and a basis set for this nullspace can be used to determine all vortex strengths $\vec{\Gamma} \in R^N$ for which the configuration remains rigid. Optimal basis sets are obtained by using the singular value decomposition of $A$ which allows one to categorize exact equilibria, approximate equilibria, and the distance between different equilibria in the appropriate vector space, as characterized by the Frobenius norm.
Gap solitons in periodic discrete nonlinear Schrödinger equations II: A generalized Nehari manifold approach
A. Pankov
2007, 19(2): 419-430 doi: 10.3934/dcds.2007.19.419 +[Abstract](494) +[PDF](171.5KB)
Abstract:
In the present article we obtain the existence of so-called ground gap solitons in discrete periodic nonlinear Schrödinger equations with cubic nonlinearity. To do that we employ a periodic approximation technique and a generalized Nehari manifold approach.
A bounded resonance problem for semilinear elliptic equations
Jiabao Su and Zhaoli Liu
2007, 19(2): 431-445 doi: 10.3934/dcds.2007.19.431 +[Abstract](579) +[PDF](196.9KB)
Abstract:
In this paper we study the existence and multiplicity of nontrivial solutions for semilinear elliptic resonance problems with a bounded nonlinearity.
On domains and their indexes with applications to semilinear elliptic equations
Hwai-Chiuan Wang
2007, 19(2): 447-467 doi: 10.3934/dcds.2007.19.447 +[Abstract](544) +[PDF](278.9KB)
Abstract:
Let $\Omega$ be a domain in $\mathbb{R}^{N}$, $N\geq1$, and $2^$∗$=\infty$ if $N=1,2$, $2^$∗$=\frac{2N}{N-2}$ if $N>2$, $2 < p < 2^$∗. Consider the semilinear elliptic equation $ -\Delta u+u=|u|^{p-2}u\text{ in }\Omega; u\in H_{0}^{1}(\Omega). $ The existence, the nonexistence, and the multiplicity of positive solutions of the equation are affected by the geometry and the topology of the domain $\Omega$. In the article, we first present various analyses and use them to characterize which domain $\Omega$ is a ground state domain or a non-ground state domain. Secondly, for a $y$-symmetric domain $\Omega$, we study their index $\alpha(\Omega)$ and $y$-symmetric index $\alpha_{s}(\Omega)$. We determine whether $\alpha(\Omega)=\alpha_{s}(\Omega)$ or $\alpha (\Omega)<\alpha_{s}(\Omega)$. In case that $\alpha(\Omega)<\alpha_{s}(\Omega)$ and that both $\alpha(\Omega)$ and $\alpha_{s}(\Omega)$ admits ground state solutions, then we obtain that in $\Omega$, the equation has three positive solutions, of which one is $y$-symmetric and other two are not $y$-symmetric.

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