# American Institute of Mathematical Sciences

June  2013, 8(2): 573-589. doi: 10.3934/nhm.2013.8.573

## Asymptotics of an optimal compliance-network problem

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

Received  October 2012 Revised  March 2013 Published  May 2013

We consider the problem of the optimal location of a Dirichlet region in a $d$-dimensional domain $\Omega$ subjected to a given force $f$ in order to minimize the $p$-compliance of the configuration. We look for the optimal region among the class of all closed connected sets of assigned length $l.$ Then we let the length $l$ tend to infinity and we look at the $\Gamma$-limit of a suitable rescaled functional, from which we get information of the asymptotic distribution of the optimal region. We also study the case where the Dirichlet region is a discrete set of finite cardinality.
Citation: 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
##### References:
 [1] G. Allaire, "Shape Optimization by the Homogenization Method," Applied Mathematical Sciences, 146, Springer-Verlag, New York 2002. [2] G. Bouchité, C.Jimenez and M. Rajesh, Asymptotique d'un problème de positionnement optimal, C. R. Acad. Sci. Paris Sér. I, 335 (2002), 1-6. [3] D. Bucur and G. Buttazzo, "Variational Methods in Shape Optimization Problems," Progress in Nonlinear Differential Equation and their Applications, 65, Birkhäuser Boston, Inc., Boston, MA, 2005. [4] D. Bucur and P. Trebeschi, Shape optimization governed by nonlinear state equations, Proc. Roy. Soc. Edinburgh Sect. A, 128 (1998), 945-963. doi: 10.1017/S0308210500030006. [5] 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. [6] 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. [7] G. Dal Maso, "An Introduction to $\Gamma$-Convergence," Progress in Nonlinear Differential Equations and their Applications, 8, Birkhäuser Boston, Inc., Boston, MA, 1993. doi: 10.1007/978-1-4612-0327-8. [8] G. Dal Maso and F. Murat, Asymptotic behavior and corrector for the Dirichlet problem in perforated domains with homogeneous monotone operators, Ann. Sc. Norm. Sup. Pisa Cl. Sci Ser. (4), 24 (1997), 239-290. [9] V. G. Maz'ja, "Sobolev Spaces," Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985. [10] S. Mosconi and P. Tilli, $\Gamma$-convergence for the irrigation problem, J. Conv. Anal., 12 (2005), 145-158. [11] V. Šverak, On optimal shape design, J. Math. Pures Appl. (9), 72 (1993), 537-551.

show all references

##### References:
 [1] G. Allaire, "Shape Optimization by the Homogenization Method," Applied Mathematical Sciences, 146, Springer-Verlag, New York 2002. [2] G. Bouchité, C.Jimenez and M. Rajesh, Asymptotique d'un problème de positionnement optimal, C. R. Acad. Sci. Paris Sér. I, 335 (2002), 1-6. [3] D. Bucur and G. Buttazzo, "Variational Methods in Shape Optimization Problems," Progress in Nonlinear Differential Equation and their Applications, 65, Birkhäuser Boston, Inc., Boston, MA, 2005. [4] D. Bucur and P. Trebeschi, Shape optimization governed by nonlinear state equations, Proc. Roy. Soc. Edinburgh Sect. A, 128 (1998), 945-963. doi: 10.1017/S0308210500030006. [5] 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. [6] 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. [7] G. Dal Maso, "An Introduction to $\Gamma$-Convergence," Progress in Nonlinear Differential Equations and their Applications, 8, Birkhäuser Boston, Inc., Boston, MA, 1993. doi: 10.1007/978-1-4612-0327-8. [8] G. Dal Maso and F. Murat, Asymptotic behavior and corrector for the Dirichlet problem in perforated domains with homogeneous monotone operators, Ann. Sc. Norm. Sup. Pisa Cl. Sci Ser. (4), 24 (1997), 239-290. [9] V. G. Maz'ja, "Sobolev Spaces," Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985. [10] S. Mosconi and P. Tilli, $\Gamma$-convergence for the irrigation problem, J. Conv. Anal., 12 (2005), 145-158. [11] V. Šverak, On optimal shape design, J. Math. Pures Appl. (9), 72 (1993), 537-551.
 [1] Jian Zhai, Zhihui Cai. $\Gamma$-convergence with Dirichlet boundary condition and Landau-Lifshitz functional for thin film. Discrete and Continuous Dynamical Systems - B, 2009, 11 (4) : 1071-1085. doi: 10.3934/dcdsb.2009.11.1071 [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] Cornel Marius Murea, Dan Tiba. Topological optimization and minimal compliance in linear elasticity. Evolution Equations and Control Theory, 2020, 9 (4) : 1115-1131. doi: 10.3934/eect.2020043 [4] 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 [5] 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 [6] Julián Fernández Bonder, Analía Silva, Juan F. Spedaletti. Gamma convergence and asymptotic behavior for eigenvalues of nonlocal problems. Discrete and Continuous Dynamical Systems, 2021, 41 (5) : 2125-2140. doi: 10.3934/dcds.2020355 [7] Gianni Dal Maso. Ennio De Giorgi and $\mathbf\Gamma$-convergence. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1017-1021. doi: 10.3934/dcds.2011.31.1017 [8] Alexander Mielke. Deriving amplitude equations via evolutionary $\Gamma$-convergence. Discrete and Continuous Dynamical Systems, 2015, 35 (6) : 2679-2700. doi: 10.3934/dcds.2015.35.2679 [9] Barbara Kaltenbacher, Gunther Peichl. The shape derivative for an optimization problem in lithotripsy. Evolution Equations and Control Theory, 2016, 5 (3) : 399-430. doi: 10.3934/eect.2016011 [10] Wenya Ma, Yihang Hao, Xiangao Liu. Shape optimization in compressible liquid crystals. Communications on Pure and Applied Analysis, 2015, 14 (5) : 1623-1639. doi: 10.3934/cpaa.2015.14.1623 [11] Lekbir Afraites, Chorouk Masnaoui, Mourad Nachaoui. Shape optimization method for an inverse geometric source problem and stability at critical shape. Discrete and Continuous Dynamical Systems - S, 2022, 15 (1) : 1-21. doi: 10.3934/dcdss.2021006 [12] Sylvia Serfaty. Gamma-convergence of gradient flows on Hilbert and metric spaces and applications. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1427-1451. doi: 10.3934/dcds.2011.31.1427 [13] Antonio De Rosa, Domenico Angelo La Manna. A non local approximation of the Gaussian perimeter: Gamma convergence and Isoperimetric properties. Communications on Pure and Applied Analysis, 2021, 20 (5) : 2101-2116. doi: 10.3934/cpaa.2021059 [14] Lorenza D'Elia. $\Gamma$-convergence of quadratic functionals with non uniformly elliptic conductivity matrices. Networks and Heterogeneous Media, 2022, 17 (1) : 15-45. doi: 10.3934/nhm.2021022 [15] Markus Muhr, Vanja Nikolić, Barbara Wohlmuth, Linus Wunderlich. Isogeometric shape optimization for nonlinear ultrasound focusing. Evolution Equations and Control Theory, 2019, 8 (1) : 163-202. doi: 10.3934/eect.2019010 [16] Benedict Geihe, Martin Rumpf. A posteriori error estimates for sequential laminates in shape optimization. Discrete and Continuous Dynamical Systems - S, 2016, 9 (5) : 1377-1392. doi: 10.3934/dcdss.2016055 [17] Günter Leugering, Jan Sokołowski, Antoni Żochowski. Control of crack propagation by shape-topological optimization. Discrete and Continuous Dynamical Systems, 2015, 35 (6) : 2625-2657. doi: 10.3934/dcds.2015.35.2625 [18] Chunlei Zhang, Qin Sheng, Raúl Ordóñez. Notes on the convergence and applications of surrogate optimization. Conference Publications, 2005, 2005 (Special) : 947-956. doi: 10.3934/proc.2005.2005.947 [19] Barbara Brandolini, Carlo Nitsch, Cristina Trombetti. Shape optimization for Monge-Ampère equations via domain derivative. Discrete and Continuous Dynamical Systems - S, 2011, 4 (4) : 825-831. doi: 10.3934/dcdss.2011.4.825 [20] Jaroslav Haslinger, Raino A. E. Mäkinen, Jan Stebel. Shape optimization for Stokes problem with threshold slip boundary conditions. Discrete and Continuous Dynamical Systems - S, 2017, 10 (6) : 1281-1301. doi: 10.3934/dcdss.2017069

2021 Impact Factor: 1.41