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 & 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.  doi: doi:10.1002/(SICI)1097-0312(199603)49:3<217::AID-CPA1>3.0.CO;2-G.  Google Scholar

[2]

B. Andrews, Contraction of convex hypersurfaces by their affine normal,, J. Differential Geom., 43 (1996), 207.   Google Scholar

[3]

B. Brandolini, C. Nitsch and C. Trombetti, New isoperimetric estimates for solutions to Monge-Ampére equations,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 26 (2009), 1265.   Google Scholar

[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.  doi: doi:10.1007/BF02871848.  Google Scholar

[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.  doi: doi:10.1002/cpa.3160370306.  Google Scholar

[6]

V. Ferone and A. Mercaldo, A second order derivation formula for functions defined by integrals,, C. R. Acad. Sci. Paris S\'er. I Math., 326 (1998), 549.   Google Scholar

[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.  doi: doi:10.1007/s00205-003-0259-4.  Google Scholar

[8]

A. Henrot and M. Pierre, "Variation et Optimisation de Formes. Une Analyse Géométrique,", Math\'ematiques & Applications, 48 (2005).   Google Scholar

[9]

C. M. Petty, Affine isoperimetric problems,, in, 440 (1985), 113.   Google Scholar

[10]

R. C. Reilly, On the Hessian of a function and the curvatures of its graph,, Michigan Math. J., 20 (1973), 373.   Google Scholar

[11]

R. Schneider, "Convex Bodies: The Brunn-Minkowski Theory,", Encyclopedia of Mathematics and its Applications, 44 (1993).   Google Scholar

[12]

J. Sokolowski and J. P. Zolésio, "Introduction to Shape Optimization. Shape Sensitivity Analysis,", Springer Series in Computational Mathematics, 16 (1992).   Google Scholar

[13]

G. Talenti, Elliptic equations and rearrangements,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 3 (1976), 697.   Google Scholar

[14]

G. Trombetti, Symmetrization methods for partial differential equations (Italian),, Boll. Un. Mat. Ital. B (8), 3 (2000), 601.   Google Scholar

show all references

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.  doi: doi:10.1002/(SICI)1097-0312(199603)49:3<217::AID-CPA1>3.0.CO;2-G.  Google Scholar

[2]

B. Andrews, Contraction of convex hypersurfaces by their affine normal,, J. Differential Geom., 43 (1996), 207.   Google Scholar

[3]

B. Brandolini, C. Nitsch and C. Trombetti, New isoperimetric estimates for solutions to Monge-Ampére equations,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 26 (2009), 1265.   Google Scholar

[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.  doi: doi:10.1007/BF02871848.  Google Scholar

[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.  doi: doi:10.1002/cpa.3160370306.  Google Scholar

[6]

V. Ferone and A. Mercaldo, A second order derivation formula for functions defined by integrals,, C. R. Acad. Sci. Paris S\'er. I Math., 326 (1998), 549.   Google Scholar

[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.  doi: doi:10.1007/s00205-003-0259-4.  Google Scholar

[8]

A. Henrot and M. Pierre, "Variation et Optimisation de Formes. Une Analyse Géométrique,", Math\'ematiques & Applications, 48 (2005).   Google Scholar

[9]

C. M. Petty, Affine isoperimetric problems,, in, 440 (1985), 113.   Google Scholar

[10]

R. C. Reilly, On the Hessian of a function and the curvatures of its graph,, Michigan Math. J., 20 (1973), 373.   Google Scholar

[11]

R. Schneider, "Convex Bodies: The Brunn-Minkowski Theory,", Encyclopedia of Mathematics and its Applications, 44 (1993).   Google Scholar

[12]

J. Sokolowski and J. P. Zolésio, "Introduction to Shape Optimization. Shape Sensitivity Analysis,", Springer Series in Computational Mathematics, 16 (1992).   Google Scholar

[13]

G. Talenti, Elliptic equations and rearrangements,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 3 (1976), 697.   Google Scholar

[14]

G. Trombetti, Symmetrization methods for partial differential equations (Italian),, Boll. Un. Mat. Ital. B (8), 3 (2000), 601.   Google Scholar

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