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

Paolo Tilli; Paolo Tilli

doi:10.3934/nhm.2012.7.127

Networks and Heterogeneous Media

2012, Volume 7, Issue 1: 127-136. doi: 10.3934/nhm.2012.7.127

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

Paolo Tilli ^{1
,}

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

Received: 01 July 2011
Primary: 74P05, 35B45; Secondary: 49J45.

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.
- Compliance,
- inequalities,
- a priori estimates
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

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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.

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