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On the Frechet differentiability in optimal control of coefficients in parabolic free boundary problems
The $\varepsilon$-entropy of some infinite dimensional compact ellipsoids and fractal dimension of attractors
| 1. | Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, P. O. Box 1160,41080-Sevilla, Spain |
| 2. | CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France |
We prove an estimation of the Kolmogorov $\varepsilon$-entropy in $H$ of the unitary ball in the space $V$, where $H$ is a Hilbert space and $V$ is a Sobolev-like subspace of $H$. Then, by means of Zelik's result [
References:
| [1] |
A. V. Babin and M. I. Vishik,
Regular attractors of semigroups and evolution equations, J. Math. Pures Appl, 62 (1983), 441-491.
|
| [2] |
Z. Chen,
A note on Kaplan-Yorke-type estimates on the fractal dimension of chaotic attractors, Chaos Solitons Fractals, 3 (1993), 575-582.
doi: 10.1016/0960-0779(93)90007-N. |
| [3] |
V. V. Chepyzhov and A. A. Ilyin,
A note on the fractal dimension of attractors of dissipative dynamical systems, Nonlinear Analysis, 44 (2001), 811-819.
doi: 10.1016/S0362-546X(99)00309-0. |
| [4] |
I. Dumer, M. S. Pinsker and V. V. Prelov,
On coverings of ellipsoids in Euclidean spaces, Transactions on Information Theory, 50 (2004), 2348-2356.
doi: 10.1109/TIT.2004.834759. |
| [5] |
P. Li and S. T. Yau,
On the Schrödinger equation and the eigenvalue problem, Comm. Math. Phys., 88 (1983), 309-318.
|
| [6] |
R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2$^{nd}$ edition, Springer-Verlag, New York, 1997.
doi: 10.1007/978-1-4612-0645-3. |
| [7] |
S. Zelik,
The attractor for a nonlinear reaction-diffusion system with a supercritical nonlinearity and its dimension, Rend. Accad. Naz. Sci. XL Mem. Mem. Math. Appl., 24 (2000), 1-25.
|
show all references
References:
| [1] |
A. V. Babin and M. I. Vishik,
Regular attractors of semigroups and evolution equations, J. Math. Pures Appl, 62 (1983), 441-491.
|
| [2] |
Z. Chen,
A note on Kaplan-Yorke-type estimates on the fractal dimension of chaotic attractors, Chaos Solitons Fractals, 3 (1993), 575-582.
doi: 10.1016/0960-0779(93)90007-N. |
| [3] |
V. V. Chepyzhov and A. A. Ilyin,
A note on the fractal dimension of attractors of dissipative dynamical systems, Nonlinear Analysis, 44 (2001), 811-819.
doi: 10.1016/S0362-546X(99)00309-0. |
| [4] |
I. Dumer, M. S. Pinsker and V. V. Prelov,
On coverings of ellipsoids in Euclidean spaces, Transactions on Information Theory, 50 (2004), 2348-2356.
doi: 10.1109/TIT.2004.834759. |
| [5] |
P. Li and S. T. Yau,
On the Schrödinger equation and the eigenvalue problem, Comm. Math. Phys., 88 (1983), 309-318.
|
| [6] |
R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2$^{nd}$ edition, Springer-Verlag, New York, 1997.
doi: 10.1007/978-1-4612-0645-3. |
| [7] |
S. Zelik,
The attractor for a nonlinear reaction-diffusion system with a supercritical nonlinearity and its dimension, Rend. Accad. Naz. Sci. XL Mem. Mem. Math. Appl., 24 (2000), 1-25.
|
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