In this paper we prove some lower bounds for the compliance
functional, in terms of the $1$-dimensional Hausdorff measure
of the Dirichlet region and the number of its connected
components. When the measure of the Dirichlet region
is large, these estimates are asymptotically optimal and
yield a proof of a conjecture by Buttazzo and Santambrogio.
Citation: Paolo Tilli. Compliance estimates for two-dimensionalproblems with Dirichlet region of prescribed length[J]. Networks and Heterogeneous Media, 2012, 7(1): 127-136. doi: 10.3934/nhm.2012.7.127
Related Papers:
| [1] |
Paolo Tilli .
Compliance estimates for two-dimensional
problems with Dirichlet region of prescribed length. Networks and Heterogeneous Media, 2012, 7(1): 127-136.
doi: 10.3934/nhm.2012.7.127
|
| [2] |
Giuseppe Buttazzo, Filippo Santambrogio .
Asymptotical compliance optimization for connected networks. Networks and Heterogeneous Media, 2007, 2(4): 761-777.
doi: 10.3934/nhm.2007.2.761
|
| [3] |
Al-hassem Nayam .
Asymptotics of an optimal compliance-network problem. Networks and Heterogeneous Media, 2013, 8(2): 573-589.
doi: 10.3934/nhm.2013.8.573
|
| [4] |
Al-hassem Nayam .
Constant in two-dimensional $p$-compliance-network problem. Networks and Heterogeneous Media, 2014, 9(1): 161-168.
doi: 10.3934/nhm.2014.9.161
|
| [5] |
Diogo A. Gomes, Gabriel E. Pires, Héctor Sánchez-Morgado .
A-priori estimates for stationary mean-field games. Networks and Heterogeneous Media, 2012, 7(2): 303-314.
doi: 10.3934/nhm.2012.7.303
|
| [6] |
Grigory Panasenko, Ruxandra Stavre .
Asymptotic analysis of the Stokes flow with variable viscosity in a thin elastic channel. Networks and Heterogeneous Media, 2010, 5(4): 783-812.
doi: 10.3934/nhm.2010.5.783
|
| [7] |
Patrick Henning, Mario Ohlberger .
The heterogeneous multiscale finite element method for advection-diffusion problems with rapidly oscillating coefficients and large expected drift. Networks and Heterogeneous Media, 2010, 5(4): 711-744.
doi: 10.3934/nhm.2010.5.711
|
| [8] |
Delio Mugnolo .
Gaussian estimates for a heat equation on a network. Networks and Heterogeneous Media, 2007, 2(1): 55-79.
doi: 10.3934/nhm.2007.2.55
|
| [9] |
Kota Kumazaki, Adrian Muntean .
Local weak solvability of a moving boundary problem describing swelling along a halfline. Networks and Heterogeneous Media, 2019, 14(3): 445-469.
doi: 10.3934/nhm.2019018
|
| [10] |
Yangyang Qiao, Huanyao Wen, Steinar Evje .
Compressible and viscous two-phase flow in porous media based on mixture theory formulation. Networks and Heterogeneous Media, 2019, 14(3): 489-536.
doi: 10.3934/nhm.2019020
|
Abstract
In this paper we prove some lower bounds for the compliance
functional, in terms of the $1$-dimensional Hausdorff measure
of the Dirichlet region and the number of its connected
components. When the measure of the Dirichlet region
is large, these estimates are asymptotically optimal and
yield a proof of a conjecture by Buttazzo and Santambrogio.
References
|
[1]
|
G. Buttazzo and F. Santambrogio, Asymptotical compliance optimization for connected networks, Netw. Heterog. Media, 2 (2007), 761-777. doi: 10.3934/nhm.2007.2.761
|
|
[2]
|
L. C. Evans and R. Gariepy, "Measure Theory and Fine Properties of Functions," Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992.
|
|
[3]
|
S. Mosconi and P. Tilli, $\Gamma$-convergence for the irrigation problem, J. Convex Anal., 12 (2005), 145-158.
|
|
[4]
|
P. Tilli, Some explicit examples of minimizers for the irrigation problem, J. Convex Anal., 17 (2010), 583-595.
|
|
[5]
|
W. P. Ziemer, "Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation," Graduate Texts in Mathematics, 120, Springer-Verlag, New York, 1989.
|
DownLoad: