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Communications on Pure & Applied Analysis

2002 , Volume 1 , Issue 3

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On the minimum time problem for driftless left-invariant control systems on SO(3)
Ugo Boscain and Yacine Chitour
2002, 1(3): 285-312 doi: 10.3934/cpaa.2002.1.285 +[Abstract](202) +[PDF](934.7KB)
Abstract:
In this paper, we investigate the structure of time-optimal trajectories for a driftless control system on $SO(3)$ of the type $\dot x=x(u_1f_1+u_2f_2), \quad |u_1|, \quad |u_2|\leq 1$, where $f_1,\quad f_2\in so(3)$ define two linearly independent left-invariant vector fields on $SO(3)$. We show that every time-optimal trajectory is a finite concatenation of at most five (bang or singular) arcs. More precisely, a time-optimal trajectory is, on the one hand, bang-bang with at most either two consecutive switchings relative to the same input or three switchings alternating between two inputs, or, on the other hand, a concatenation of at most two bangs followed by a singular arc and then two other bangs. We end up finding a finite number of three-parameters trajectory types that are sufficient for time-optimality.
The evolution thermistor problem with degenerate thermal conductivity
María Teresa González Montesinos and Francisco Ortegón Gallego
2002, 1(3): 313-325 doi: 10.3934/cpaa.2002.1.313 +[Abstract](267) +[PDF](217.4KB)
Abstract:
The existence of a weak solution for the time dependent thermistor problem with degenerate thermal conductivity is proved in this work. The main difficulties of this problem lies on the absence of space estimates for the temperature and time estimates for the electrical potential.
The global minimizers and vortex solutions to a Ginzburg-Landau model of superconducting films
Shijin Ding and Qiang Du
2002, 1(3): 327-340 doi: 10.3934/cpaa.2002.1.327 +[Abstract](260) +[PDF](217.6KB)
Abstract:
In this paper, we discuss the global minimizers of a free energy for the superconducting thin films placed in a magnetic field $h_{e x}$ below the lower critical field $H_{c1}$ or between $H_{c1}$ and the upper critical field $H_{c2}$. For $h_{e x}$ is near but smaller than $H_{c1}$, we prove that the global minimizer having no vortex is unique. For $H_{c1}$<<$h_{e x}$<<$H_{c2}$, we prove that the density of the vortices of the global minimizer is proportional to the applied field.
An adaptive mesh redistribution algorithm for convection-dominated problems
Zheng-Ru Zhang and Tao Tang
2002, 1(3): 341-357 doi: 10.3934/cpaa.2002.1.341 +[Abstract](285) +[PDF](1306.2KB)
Abstract:
Convection-dominated problems are of practical applications and in general may require extremely fine meshes over a small portion of the physical domain. In this work an efficient adaptive mesh redistribution (AMR) algorithm will be developed for solving one- and two-dimensional convection-dominated problems. Several test problems are computed by using the proposed algorithm. The adaptive mesh results are compared with those obtained with uniform meshes to demonstrate the effectiveness and robustness of the proposed algorithm.
Asymptotic behavior of the best Sobolev trace constant in expanding and contracting domains
Julián Fernández Bonder and Julio D. Rossi
2002, 1(3): 359-378 doi: 10.3934/cpaa.2002.1.359 +[Abstract](305) +[PDF](239.6KB)
Abstract:
We study the asymptotic behavior for the best constant and extremals of the Sobolev trace embedding $W^{1,p} (\Omega) \rightarrow L^q (\partial \Omega)$ on expanding and contracting domains. We find that the behavior strongly depends on $p$ and $q$. For contracting domains we prove that the behavior of the best Sobolev trace constant depends on the sign of $qN-pN+p$ while for expanding domains it depends on the sign of $q-p$. We also give some results regarding the behavior of the extremals, for contracting domains we prove that they converge to a constant when rescaled in a suitable way and for expanding domains we observe when a concentration phenomena takes place.
In a horizontal layer with free upper surface
Bum Ja Jin and Mariarosaria Padula
2002, 1(3): 379-415 doi: 10.3934/cpaa.2002.1.379 +[Abstract](231) +[PDF](345.9KB)
Abstract:
We propose a new existence proof of global in time solutions of isothermal viscous gases in a layer bounded below by a horizontal plane, and above by a free upper surface, which are periodic in the two horizontal variables. Despite the importance of compressible fluids for physical applications, the problem of uniform in time estimates is scarcely explored. The rest state with a steady distribution of density in a rectangular domain is stable, without restrictions on initial data, in a "weak" norm provided the flows exist in a suitable regularity class. In this paper we show existence of regular global in time solutions, and the exponential decay of these solutions to the rest as time goes to $\infty$, when the initial data are small perturbation of the basic flow. The analysis presented here is based on estimates in Hilbert spaces.
On existence and concentration behavior of ground state solutions for a class of problems with critical growth
Claudianor Oliveira Alves and M. A.S. Souto
2002, 1(3): 417-431 doi: 10.3934/cpaa.2002.1.417 +[Abstract](309) +[PDF](235.8KB)
Abstract:
In this paper, we study the existence and the concentration behavior of ground state for the problem

$-h^2\Delta u+V(z)u=\lambda u^q+u^{2^{ *} -1,\mathbb R^N $

$u(z)>0\quad $ for all $z\in \mathbb R^N \qquad\qquad\qquad\qquad\qquad\qquad\qquad (P_{h})$

where $h, \lambda >0$, 1<$q$ <$2^{ * -1$ $=\frac{N+2}{N-2}$, $N\geq 3$ and $V: \mathbb R^N\to \mathbb R$ is a positive function such that

0< $i nf_{z\in\mathbb R^N}V(z)$< $limi nf_{|z| \rightarrow \infty}V(z)=V_{\infty}.$

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