ISSN:
1534-0392
eISSN:
1553-5258
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Communications on Pure & Applied Analysis
September 2002 , Volume 1 , Issue 3
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2002, 1(3): 285-312
doi: 10.3934/cpaa.2002.1.285
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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.
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.
2002, 1(3): 313-325
doi: 10.3934/cpaa.2002.1.313
+[Abstract](1166)
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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 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.
2002, 1(3): 327-340
doi: 10.3934/cpaa.2002.1.327
+[Abstract](1174)
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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.
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.
2002, 1(3): 341-357
doi: 10.3934/cpaa.2002.1.341
+[Abstract](1241)
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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.
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.
2002, 1(3): 359-378
doi: 10.3934/cpaa.2002.1.359
+[Abstract](1335)
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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.
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.
2002, 1(3): 379-415
doi: 10.3934/cpaa.2002.1.379
+[Abstract](1218)
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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.
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.
2002, 1(3): 417-431
doi: 10.3934/cpaa.2002.1.417
+[Abstract](1567)
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Abstract:
In this paper, we study the existence and the concentration behavior of ground state for the problem
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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