American Institute of Mathematical Sciences

August  2011, 4(4): 825-831. doi: 10.3934/dcdss.2011.4.825

Shape optimization for Monge-Ampère equations via domain derivative

 1 Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli Studi di Napoli “Federico II”, Complesso Universitario Monte S. Angelo, via Cintia, 80126 Napoli 2 Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli Studi di Napoli “Federico II”, Via Cintia, Monte S. Angelo, I-80126 Napoli

Received  October 2009 Revised  January 2010 Published  November 2010

In this note we prove that, if $\Omega$ is a smooth, strictly convex, open set in $R^n$ $(n \ge 2)$ with given measure, the $L^1$ norm of the convex solution to the Dirichlet problem $\det D^2 u=1$ in $\Omega$, $u=0$ on $\partial\Omega$, is minimum whenever $\Omega$ is an ellipsoid.
Citation: 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
References:
 [1] A. Alvino, J. I. Diaz, P. L. Lions and G. Trombetti, Elliptic equations and Steiner symmetrization, Comm. Pure Appl. Math., 49 (1996), 217-236. doi: doi:10.1002/(SICI)1097-0312(199603)49:3<217::AID-CPA1>3.0.CO;2-G. [2] B. Andrews, Contraction of convex hypersurfaces by their affine normal, J. Differential Geom., 43 (1996), 207-230. [3] B. Brandolini, C. Nitsch and C. Trombetti, New isoperimetric estimates for solutions to Monge-Ampére equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1265-1275. [4] F. Brock and A. Henrot, A symmetry result for an overdetermined elliptic problem using continuous rearrangement and domain derivative, Rend. Circ. Mat. Palermo (2), 51 (2002), 375-390. doi: doi:10.1007/BF02871848. [5] L. Caffarelli, L. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations. I. Monge-Amp\ere equation, Comm. Pure Appl. Math., 37 (1984), 369-402. doi: doi:10.1002/cpa.3160370306. [6] V. Ferone and A. Mercaldo, A second order derivation formula for functions defined by integrals, C. R. Acad. Sci. Paris Sér. I Math., 326 (1998), 549-554. [7] A. Henrot and E. Oudet, Minimizing the second eigenvalue of the Laplace operator with Dirichlet boundary conditions, Arch. Ration. Mech. Anal., 169 (2003), 73-87. doi: doi:10.1007/s00205-003-0259-4. [8] A. Henrot and M. Pierre, "Variation et Optimisation de Formes. Une Analyse Géométrique," Mathématiques & Applications, vol. 48, Springer, 2005. [9] C. M. Petty, Affine isoperimetric problems, in "Discrete Geometry and Convexity" (New York, 1982), Ann. New York Acad. Sci., vol. 440, New York Acad. Sci., New York, (1985), 113-127. [10] R. C. Reilly, On the Hessian of a function and the curvatures of its graph, Michigan Math. J., 20 (1973), 373-383. [11] R. Schneider, "Convex Bodies: The Brunn-Minkowski Theory," Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1993. [12] J. Sokolowski and J. P. Zolésio, "Introduction to Shape Optimization. Shape Sensitivity Analysis," Springer Series in Computational Mathematics, vol. 16, Springer-Verlag, Berlin, 1992. [13] G. Talenti, Elliptic equations and rearrangements, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 3 (1976), 697-718. [14] G. Trombetti, Symmetrization methods for partial differential equations (Italian), Boll. Un. Mat. Ital. B (8), 3 (2000), 601-634.

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References:
 [1] A. Alvino, J. I. Diaz, P. L. Lions and G. Trombetti, Elliptic equations and Steiner symmetrization, Comm. Pure Appl. Math., 49 (1996), 217-236. doi: doi:10.1002/(SICI)1097-0312(199603)49:3<217::AID-CPA1>3.0.CO;2-G. [2] B. Andrews, Contraction of convex hypersurfaces by their affine normal, J. Differential Geom., 43 (1996), 207-230. [3] B. Brandolini, C. Nitsch and C. Trombetti, New isoperimetric estimates for solutions to Monge-Ampére equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1265-1275. [4] F. Brock and A. Henrot, A symmetry result for an overdetermined elliptic problem using continuous rearrangement and domain derivative, Rend. Circ. Mat. Palermo (2), 51 (2002), 375-390. doi: doi:10.1007/BF02871848. [5] L. Caffarelli, L. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations. I. Monge-Amp\ere equation, Comm. Pure Appl. Math., 37 (1984), 369-402. doi: doi:10.1002/cpa.3160370306. [6] V. Ferone and A. Mercaldo, A second order derivation formula for functions defined by integrals, C. R. Acad. Sci. Paris Sér. I Math., 326 (1998), 549-554. [7] A. Henrot and E. Oudet, Minimizing the second eigenvalue of the Laplace operator with Dirichlet boundary conditions, Arch. Ration. Mech. Anal., 169 (2003), 73-87. doi: doi:10.1007/s00205-003-0259-4. [8] A. Henrot and M. Pierre, "Variation et Optimisation de Formes. Une Analyse Géométrique," Mathématiques & Applications, vol. 48, Springer, 2005. [9] C. M. Petty, Affine isoperimetric problems, in "Discrete Geometry and Convexity" (New York, 1982), Ann. New York Acad. Sci., vol. 440, New York Acad. Sci., New York, (1985), 113-127. [10] R. C. Reilly, On the Hessian of a function and the curvatures of its graph, Michigan Math. J., 20 (1973), 373-383. [11] R. Schneider, "Convex Bodies: The Brunn-Minkowski Theory," Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1993. [12] J. Sokolowski and J. P. Zolésio, "Introduction to Shape Optimization. Shape Sensitivity Analysis," Springer Series in Computational Mathematics, vol. 16, Springer-Verlag, Berlin, 1992. [13] G. Talenti, Elliptic equations and rearrangements, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 3 (1976), 697-718. [14] G. Trombetti, Symmetrization methods for partial differential equations (Italian), Boll. Un. Mat. Ital. B (8), 3 (2000), 601-634.
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