# American Institute of Mathematical Sciences

March  2014, 9(1): 161-168. doi: 10.3934/nhm.2014.9.161

## Constant in two-dimensional $p$-compliance-network problem

 1 International Centre for Theoretical Physics, Strada Costiera,11, I - 34151 Trieste

Received  April 2013 Revised  October 2013 Published  April 2014

We consider the problem of the minimization of the $p$-compliance functional where the control variables $\Sigma$ are taking among closed connected one-dimensional sets. We prove some estimate from below of the $p$-compliance functional in terms of the one-dimensional Hausdorff measure of $\Sigma$ and compute the value of a constant $\theta(p)$ appearing usually in $\Gamma$-limit functional of the rescaled $p$-compliance functional.
Citation: 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
##### References:
 [1] D. Bucur and P. Trebeschi, Shape optimization governed by nonlinear state equations, Proc. Roy. Soc. Edinburgh - A, 128 (1998), 945-963. doi: 10.1017/S0308210500030006. [2] G. Buttazzo and F. Santambrogio, Asymptotical compliance optimization for connected networks, Networks and Heterogeneous Media, 2 (2007), 761-777. doi: 10.3934/nhm.2007.2.761. [3] G. Buttazzo, F. Santambrogio and N. Varchon, Asymptotics of an optimal compliance-location problem, ESAIM Control Optimization and Calculus of Variations, 12 (2006), 752-769. doi: 10.1051/cocv:2006020. [4] G. Dal Maso, An Introduction to $\Gamma$-Convergence, Birkhäuser, Basel, 1993. doi: 10.1007/978-1-4612-0327-8. [5] S. Mosconi and P. Tilli, $\Gamma$-convergence for the irrigation problem, J. Conv. Anal., 12 (2005), 145-158. [6] P. Tilli, Compliance estimates for two-dimensional problems with Dirichlet region of prescribed length, Networks and Heterogeneous Media, 7 (2012), 127-136. doi: 10.3934/nhm.2012.7.127. [7] W. P. Ziemer, Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation, Graduate texts in Mathematics, 120, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4612-1015-3.

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##### References:
 [1] D. Bucur and P. Trebeschi, Shape optimization governed by nonlinear state equations, Proc. Roy. Soc. Edinburgh - A, 128 (1998), 945-963. doi: 10.1017/S0308210500030006. [2] G. Buttazzo and F. Santambrogio, Asymptotical compliance optimization for connected networks, Networks and Heterogeneous Media, 2 (2007), 761-777. doi: 10.3934/nhm.2007.2.761. [3] G. Buttazzo, F. Santambrogio and N. Varchon, Asymptotics of an optimal compliance-location problem, ESAIM Control Optimization and Calculus of Variations, 12 (2006), 752-769. doi: 10.1051/cocv:2006020. [4] G. Dal Maso, An Introduction to $\Gamma$-Convergence, Birkhäuser, Basel, 1993. doi: 10.1007/978-1-4612-0327-8. [5] S. Mosconi and P. Tilli, $\Gamma$-convergence for the irrigation problem, J. Conv. Anal., 12 (2005), 145-158. [6] P. Tilli, Compliance estimates for two-dimensional problems with Dirichlet region of prescribed length, Networks and Heterogeneous Media, 7 (2012), 127-136. doi: 10.3934/nhm.2012.7.127. [7] W. P. Ziemer, Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation, Graduate texts in Mathematics, 120, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4612-1015-3.
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