# American Institute of Mathematical Sciences

December  2011, 31(4): 1115-1128. doi: 10.3934/dcds.2011.31.1115

## Optimal shape for elliptic problems with random perturbations

 1 Dipartimento di Matematica, Largo B. Pontecorvo, 5, 56127 Pisa, Italy 2 Departamento de Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, Tarﬁa s/n, E-41012 Sevilla, Spain

Received  October 2009 Revised  March 2010 Published  September 2011

In this paper we analyze the relaxed form of a shape optimization problem with state equation $$$$\left\{\begin{array}{ll} -div\big(a(x)Du\big)=f\qquad\hbox{in }D\\ \hbox{boundary conditions on }\partial D. \end{array} \right.$$$$ The new fact is that the term $f$ is only known up to a random perturbation $\xi(x,\omega)$. The goal is to find an optimal coefficient $a(x)$, fulfilling the usual constraints $\alpha\le a\le\beta$ and $\displaystyle\int_D a(x)\,dx\le m$, which minimizes a cost function of the form $$\int_\Omega\int_Dj\big(x,\omega,u_a(x,\omega)\big)\,dx\,dP(\omega).$$ Some numerical examples are shown in the last section, which evidence the previous analytical results.
Citation: Giuseppe Buttazzo, Faustino Maestre. Optimal shape for elliptic problems with random perturbations. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1115-1128. doi: 10.3934/dcds.2011.31.1115
##### References:
 [1] G. Allaire, "Shape Optimization by the Homogenization Method," Applied Mathematical Sciences, 146, Springer-Verlag, New York, 2002. [2] F. Alvarez and M. Carrasco, Minimization of the expected compliance as an alternative approach to multiload truss optimization, Struct. Multidiscip. Optim., 29 (2005), 470-476. doi: 10.1007/s00158-004-0488-7. [3] M. P. Bendsøe and O. Sigmund, "Topology Optimization: Theory, Methods and Aplications," Springer-Verlag, Berlin, 2003. [4] D. Bucur and G. Buttazzo, "Variational Methods in Shape Optimization Problems," Progress in Nonlinear Differential Equations and their Applications, 65, Birkhäuser Boston, Inc., Boston, MA, 2005. [5] J. Casado-Díaz, J. Couce-Calvo, M. Luna-Laynez and J. D. Martín-Gómez, Optimal design problems for a non-linear cost in the gradient: numerical results, Applicable Analysis, 87 (2008), 1461-1487. [6] A. Cherkaev, "Variational Methods for Structural Optimization," Applied Mathematical Sciences, 140, Springer-Verlag, New York, 2000. [7] G. Dal Maso and L. Modica, Nonlinear stochastic homogenization and ergodic theory, J. Reine Angew. Math., 368 (1986), 28-42. [8] A. Donoso and P. Pedregal, Optimal design of 2-$D$ conducting graded materials by minimizing quadratic funtionals in the field, Struc. Multidisc Optim., 30 (2005), 360-367. doi: 10.1007/s00158-005-0521-5. [9] A. Henrot and M. Pierre, "Variation et Optimisation de Formes. Une Analyse Géométrique," Mathématiques et Applications, 48, Springer, Berlin, 2005. [10] K. A. Lurie and A. V. Cherkaev, Exact estimates of conductivity of composites formed by two isotropically conducting media taken in prescribed proportion, Proc. R. Soc. Edinb. A, 99 (1984), 71-87. [11] F. Maestre, A. Münch and P. Pedregal, A spatio-temporal design problem for a damped wave equation, SIAM J. Appl. Math., 68 (2007), 109-132. doi: 10.1137/07067965X. [12] G. W. Milton, "The Theory of Composites," Cambridge Monographs on Applied and Computational Mathematics, 6, Cambridge University Press, Cambridge, 2002. [13] F. Murat, Contre-exemples pour divers problèmes où le contrôle intervient dans les coefficients, Ann. Mat. Pura Appl. (4), 112 (1977), 49-68. [14] F. Murat and L. Tartar, Optimality conditions and homogenization, in "Nonlinear Variational Problems" (Isola d'Elba, 1983), Res. Notes Math., 127, Pitman, Boston, MA, (1985), 1-8. [15] F. Murat and L. Tartar, $H$-convergence, in "Topics in the Mathematical Modelling of Composites Materials," Progress in Nonlinear Differential Equations, 31, Birkhäuser Boston, Boston, MA, (1997), 21-43. [16] F. Murat and L. Tartar, Calculus of variations and homogenization, in "Topics in the Mathematical Modelling of Composites Materials," Progress in Nonlinear Differential Equations Appl., 31, Birkhäuser Boston, Boston, MA, (1997), 139-173. [17] P. Pedregal, Vector variational problems and applications to optimal design, ESAIM Control Optim. Calc. Var., 11 (2005), 357-381. doi: 10.1051/cocv:2005010. [18] P. Pedregal, "Parametrized Measures and Variational Principles," Progress Nonlinear Differential Equations and Their Applications, 30, Birkhäuser Verlag, Basel, 1997. [19] J. Sokołowski and J.-P. Zolésio, "Introduction to Shape Optimization: Shape Sensitivity Analysis," Springer Series in Computational Mathematics, 16, Springer-Verlag, Berlin, 1992. [20] L. Tartar, An introduction to the homogenization method in optimal design, in "Optimal Shape Design" (Tróia, 1998), Lecture Notes Math., 1740, Springer, Berlin, (2000), 47-156.

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##### References:
 [1] G. Allaire, "Shape Optimization by the Homogenization Method," Applied Mathematical Sciences, 146, Springer-Verlag, New York, 2002. [2] F. Alvarez and M. Carrasco, Minimization of the expected compliance as an alternative approach to multiload truss optimization, Struct. Multidiscip. Optim., 29 (2005), 470-476. doi: 10.1007/s00158-004-0488-7. [3] M. P. Bendsøe and O. Sigmund, "Topology Optimization: Theory, Methods and Aplications," Springer-Verlag, Berlin, 2003. [4] D. Bucur and G. Buttazzo, "Variational Methods in Shape Optimization Problems," Progress in Nonlinear Differential Equations and their Applications, 65, Birkhäuser Boston, Inc., Boston, MA, 2005. [5] J. Casado-Díaz, J. Couce-Calvo, M. Luna-Laynez and J. D. Martín-Gómez, Optimal design problems for a non-linear cost in the gradient: numerical results, Applicable Analysis, 87 (2008), 1461-1487. [6] A. Cherkaev, "Variational Methods for Structural Optimization," Applied Mathematical Sciences, 140, Springer-Verlag, New York, 2000. [7] G. Dal Maso and L. Modica, Nonlinear stochastic homogenization and ergodic theory, J. Reine Angew. Math., 368 (1986), 28-42. [8] A. Donoso and P. Pedregal, Optimal design of 2-$D$ conducting graded materials by minimizing quadratic funtionals in the field, Struc. Multidisc Optim., 30 (2005), 360-367. doi: 10.1007/s00158-005-0521-5. [9] A. Henrot and M. Pierre, "Variation et Optimisation de Formes. Une Analyse Géométrique," Mathématiques et Applications, 48, Springer, Berlin, 2005. [10] K. A. Lurie and A. V. Cherkaev, Exact estimates of conductivity of composites formed by two isotropically conducting media taken in prescribed proportion, Proc. R. Soc. Edinb. A, 99 (1984), 71-87. [11] F. Maestre, A. Münch and P. Pedregal, A spatio-temporal design problem for a damped wave equation, SIAM J. Appl. Math., 68 (2007), 109-132. doi: 10.1137/07067965X. [12] G. W. Milton, "The Theory of Composites," Cambridge Monographs on Applied and Computational Mathematics, 6, Cambridge University Press, Cambridge, 2002. [13] F. Murat, Contre-exemples pour divers problèmes où le contrôle intervient dans les coefficients, Ann. Mat. Pura Appl. (4), 112 (1977), 49-68. [14] F. Murat and L. Tartar, Optimality conditions and homogenization, in "Nonlinear Variational Problems" (Isola d'Elba, 1983), Res. Notes Math., 127, Pitman, Boston, MA, (1985), 1-8. [15] F. Murat and L. Tartar, $H$-convergence, in "Topics in the Mathematical Modelling of Composites Materials," Progress in Nonlinear Differential Equations, 31, Birkhäuser Boston, Boston, MA, (1997), 21-43. [16] F. Murat and L. Tartar, Calculus of variations and homogenization, in "Topics in the Mathematical Modelling of Composites Materials," Progress in Nonlinear Differential Equations Appl., 31, Birkhäuser Boston, Boston, MA, (1997), 139-173. [17] P. Pedregal, Vector variational problems and applications to optimal design, ESAIM Control Optim. Calc. Var., 11 (2005), 357-381. doi: 10.1051/cocv:2005010. [18] P. Pedregal, "Parametrized Measures and Variational Principles," Progress Nonlinear Differential Equations and Their Applications, 30, Birkhäuser Verlag, Basel, 1997. [19] J. Sokołowski and J.-P. Zolésio, "Introduction to Shape Optimization: Shape Sensitivity Analysis," Springer Series in Computational Mathematics, 16, Springer-Verlag, Berlin, 1992. [20] L. Tartar, An introduction to the homogenization method in optimal design, in "Optimal Shape Design" (Tróia, 1998), Lecture Notes Math., 1740, Springer, Berlin, (2000), 47-156.
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