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), 217-236. 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), 207-230.  Google Scholar

[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.  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), 375-390. 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), 369-402. 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 Sér. I Math., 326 (1998), 549-554.  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), 73-87. doi: doi:10.1007/s00205-003-0259-4.  Google Scholar

[8]

A. Henrot and M. Pierre, "Variation et Optimisation de Formes. Une Analyse Géométrique," Mathématiques & Applications, vol. 48, Springer, 2005.  Google Scholar

[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.  Google Scholar

[10]

R. C. Reilly, On the Hessian of a function and the curvatures of its graph, Michigan Math. J., 20 (1973), 373-383.  Google Scholar

[11]

R. Schneider, "Convex Bodies: The Brunn-Minkowski Theory," Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1993.  Google Scholar

[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.  Google Scholar

[13]

G. Talenti, Elliptic equations and rearrangements, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 3 (1976), 697-718.  Google Scholar

[14]

G. Trombetti, Symmetrization methods for partial differential equations (Italian), Boll. Un. Mat. Ital. B (8), 3 (2000), 601-634.  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), 217-236. 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), 207-230.  Google Scholar

[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.  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), 375-390. 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), 369-402. 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 Sér. I Math., 326 (1998), 549-554.  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), 73-87. doi: doi:10.1007/s00205-003-0259-4.  Google Scholar

[8]

A. Henrot and M. Pierre, "Variation et Optimisation de Formes. Une Analyse Géométrique," Mathématiques & Applications, vol. 48, Springer, 2005.  Google Scholar

[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.  Google Scholar

[10]

R. C. Reilly, On the Hessian of a function and the curvatures of its graph, Michigan Math. J., 20 (1973), 373-383.  Google Scholar

[11]

R. Schneider, "Convex Bodies: The Brunn-Minkowski Theory," Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1993.  Google Scholar

[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.  Google Scholar

[13]

G. Talenti, Elliptic equations and rearrangements, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 3 (1976), 697-718.  Google Scholar

[14]

G. Trombetti, Symmetrization methods for partial differential equations (Italian), Boll. Un. Mat. Ital. B (8), 3 (2000), 601-634.  Google Scholar

[1]

Qi-Rui Li, Xu-Jia Wang. Regularity of the homogeneous Monge-Ampère equation. Discrete & Continuous Dynamical Systems, 2015, 35 (12) : 6069-6084. doi: 10.3934/dcds.2015.35.6069

[2]

Luca Codenotti, Marta Lewicka. Visualization of the convex integration solutions to the Monge-Ampère equation. Evolution Equations & Control Theory, 2019, 8 (2) : 273-300. doi: 10.3934/eect.2019015

[3]

Bo Guan, Qun Li. A Monge-Ampère type fully nonlinear equation on Hermitian manifolds. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1991-1999. doi: 10.3934/dcdsb.2012.17.1991

[4]

Alessio Figalli, Young-Heon Kim. Partial regularity of Brenier solutions of the Monge-Ampère equation. Discrete & Continuous Dynamical Systems, 2010, 28 (2) : 559-565. doi: 10.3934/dcds.2010.28.559

[5]

Adam M. Oberman. Wide stencil finite difference schemes for the elliptic Monge-Ampère equation and functions of the eigenvalues of the Hessian. Discrete & Continuous Dynamical Systems - B, 2008, 10 (1) : 221-238. doi: 10.3934/dcdsb.2008.10.221

[6]

Zhijun Zhang. Optimal global asymptotic behavior of the solution to a singular monge-ampère equation. Communications on Pure & Applied Analysis, 2020, 19 (2) : 1129-1145. doi: 10.3934/cpaa.2020053

[7]

Shuyu Gong, Ziwei Zhou, Jiguang Bao. Existence and uniqueness of viscosity solutions to the exterior problem of a parabolic Monge-Ampère equation. Communications on Pure & Applied Analysis, 2020, 19 (10) : 4921-4936. doi: 10.3934/cpaa.2020218

[8]

Diego Maldonado. On interior $C^2$-estimates for the Monge-Ampère equation. Discrete & Continuous Dynamical Systems, 2018, 38 (3) : 1427-1440. doi: 10.3934/dcds.2018058

[9]

Yahui Niu. Monotonicity of solutions for a class of nonlocal Monge-Ampère problem. Communications on Pure & Applied Analysis, 2020, 19 (11) : 5269-5283. doi: 10.3934/cpaa.2020237

[10]

Limei Dai, Hongyu Li. Entire subsolutions of Monge-Ampère type equations. Communications on Pure & Applied Analysis, 2020, 19 (1) : 19-30. doi: 10.3934/cpaa.2020002

[11]

Jiakun Liu, Neil S. Trudinger. On Pogorelov estimates for Monge-Ampère type equations. Discrete & Continuous Dynamical Systems, 2010, 28 (3) : 1121-1135. doi: 10.3934/dcds.2010.28.1121

[12]

Fan Cui, Huaiyu Jian. Symmetry of solutions to a class of Monge-Ampère equations. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1247-1259. doi: 10.3934/cpaa.2019060

[13]

Juhua Shi, Feida Jiang. The degenerate Monge-Ampère equations with the Neumann condition. Communications on Pure & Applied Analysis, 2021, 20 (2) : 915-931. doi: 10.3934/cpaa.2020297

[14]

Limei Dai. Multi-valued solutions to a class of parabolic Monge-Ampère equations. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1061-1074. doi: 10.3934/cpaa.2014.13.1061

[15]

Mengni Li. Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions. Communications on Pure & Applied Analysis, 2021, 20 (1) : 301-317. doi: 10.3934/cpaa.2020267

[16]

Jingang Xiong, Jiguang Bao. The obstacle problem for Monge-Ampère type equations in non-convex domains. Communications on Pure & Applied Analysis, 2011, 10 (1) : 59-68. doi: 10.3934/cpaa.2011.10.59

[17]

Cristian Enache. Maximum and minimum principles for a class of Monge-Ampère equations in the plane, with applications to surfaces of constant Gauss curvature. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1347-1359. doi: 10.3934/cpaa.2014.13.1347

[18]

Shouchuan Hu, Haiyan Wang. Convex solutions of boundary value problem arising from Monge-Ampère equations. Discrete & Continuous Dynamical Systems, 2006, 16 (3) : 705-720. doi: 10.3934/dcds.2006.16.705

[19]

Ziwei Zhou, Jiguang Bao, Bo Wang. A Liouville theorem of parabolic Monge-AmpÈre equations in half-space. Discrete & Continuous Dynamical Systems, 2021, 41 (4) : 1561-1578. doi: 10.3934/dcds.2020331

[20]

Haitao Yang, Yibin Chang. On the blow-up boundary solutions of the Monge -Ampére equation with singular weights. Communications on Pure & Applied Analysis, 2012, 11 (2) : 697-708. doi: 10.3934/cpaa.2012.11.697

2020 Impact Factor: 2.425

Metrics

  • PDF downloads (64)
  • HTML views (0)
  • Cited by (1)

[Back to Top]