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March  2012, 7(1): 127-136. doi: 10.3934/nhm.2012.7.127

## Compliance estimates for two-dimensional problems with Dirichlet region of prescribed length

 1 Politecnico di Torino, Dipartimento di Scienze Matematiche, Corso Duca degli Abruzzi, 24 10129 Torino, Italy

Received  July 2011 Published  February 2012

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-dimensional problems with Dirichlet region of prescribed length. Networks and Heterogeneous Media, 2012, 7 (1) : 127-136. doi: 10.3934/nhm.2012.7.127
##### 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.

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##### 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.
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