November  2014, 13(6): 2305-2316. doi: 10.3934/cpaa.2014.13.2305

Mirror symmetry for a Hessian over-determined problem and its generalization

1. 

School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, China

Received  September 2013 Revised  February 2014 Published  July 2014

In the paper, we apply the moving plane method to prove that if the right hand sides of equation and Neumann boundary condition are both independent of one variable, the domain and the solution to the Hessian over-determined problem are mirror symmetric. Our result generalizes the previous results on radial symmetry. In the end, we get the mirror symmetry of over-determined problems for more general equations, which include Weingarten curvature equation.
Citation: Bo Wang, Jiguang Bao. Mirror symmetry for a Hessian over-determined problem and its generalization. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2305-2316. doi: 10.3934/cpaa.2014.13.2305
References:
[1]

B. Brandolini, C. Nitsch, P. Salani and C. Trombetti, Serrin-type overdetermined problems: An alternative proof,, \emph{Arch. Rational Mech. Anal.}, 190 (2008), 267.  doi: 10.1007/s00205-008-0119-3.  Google Scholar

[2]

B. Brandolini, C. Nitsch, P. Salani and C. Trombetti, On the stability of the Serrin problem,, \emph{J. Differ. Equ.}, 245 (2008), 1566.  doi: 10.1016/j.jde.2008.06.010.  Google Scholar

[3]

B. Brandolini, C. Nitsch, P. Salani and C. Trombetti, Stability of radial symmetry for a Monge-Ampere overdetermined problem,, \emph{Annali di Matematica.}, 188 (2009), 445.  doi: 10.1007/s10231-008-0083-4.  Google Scholar

[4]

L. Caffarelli, L. Nirenberg and J. Spruck, Nonlinear second order equations IV. Starshaped compact Weingurtenhypersurfuces,, \emph{Current Topics Inpurtiul Differential Equations}, (1986), 1.   Google Scholar

[5]

L. Caffarelli, L. Nirenberg and J. Spruck, Nonlinear second order elliptic equations V. The Dirichlet problem for Weingurten hypersurfuces,, \emph{Communications on Pure and Applied Mathematics}, 41 (1988), 47.  doi: 10.1002/cpa.3160410105.  Google Scholar

[6]

L. E. Fraenkel, An introduction to Maximum Principles and Symmetry in Elliptic Problems,, Cambridge Tracts in Mathematics, (2000).  doi: 10.1017/CBO9780511569203.  Google Scholar

[7]

B. Gidas, Weiming Ni and L. Nirenberg, Symmetry and related properties via the maximum principle,, \emph{Comm. Math. Phys.}, 68 (1979), 209.   Google Scholar

[8]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Second edition, (1983).  doi: 10.1007/978-3-642-61798-0.  Google Scholar

[9]

B. Kawohl, Symmetrzationor how to prove symmetry of solutions to a PDE,, \emph{Partial differential equations} (Praha), (1998), 214.   Google Scholar

[10]

Congming Li, Monotonicity and symmetry of solutions of fully nonlinear elliptic equations on unbounded domains,, \emph{Comm. Partial Differential Equations}, 16 (1991), 585.  doi: 10.1080/03605309108820770.  Google Scholar

[11]

W. Reichel, Radial symmetry for elliptic boundary-value problems on exterior domains,, \emph{Arch. Rational Mech. Anal.}, 137 (1997), 381.  doi: 10.1007/s002050050034.  Google Scholar

[12]

W. Reichel, Radial symmetry for an electrostatic, a capillarity and some fully nonlinear overdetermined problems on exterior domains,, \emph{Z. Anal. Anwendungen, 15 (1996), 619.  doi: 10.4171/ZAA/719.  Google Scholar

[13]

J. Serrin, A symmetry problem in potential theory,, \emph{Arch. Rational Mech. Anal.}, 43 (1971), 304.   Google Scholar

[14]

H. Shahgholian, Diversifications of Serrin's and related symmetry problems,, \emph{Complex Variables and Elliptic Equations, 57 (2012), 653.  doi: 10.1080/17476933.2010.504848.  Google Scholar

[15]

I. S. Sokolnikoff, Mathematical Theory of Elasticity,, McGraw-Hill, (1956).   Google Scholar

[16]

H. F. Weinberger, Remark on the preceding paper of Serrin,, \emph{Arch. Rational Mech. Anal.}, 43 (1971), 319.   Google Scholar

show all references

References:
[1]

B. Brandolini, C. Nitsch, P. Salani and C. Trombetti, Serrin-type overdetermined problems: An alternative proof,, \emph{Arch. Rational Mech. Anal.}, 190 (2008), 267.  doi: 10.1007/s00205-008-0119-3.  Google Scholar

[2]

B. Brandolini, C. Nitsch, P. Salani and C. Trombetti, On the stability of the Serrin problem,, \emph{J. Differ. Equ.}, 245 (2008), 1566.  doi: 10.1016/j.jde.2008.06.010.  Google Scholar

[3]

B. Brandolini, C. Nitsch, P. Salani and C. Trombetti, Stability of radial symmetry for a Monge-Ampere overdetermined problem,, \emph{Annali di Matematica.}, 188 (2009), 445.  doi: 10.1007/s10231-008-0083-4.  Google Scholar

[4]

L. Caffarelli, L. Nirenberg and J. Spruck, Nonlinear second order equations IV. Starshaped compact Weingurtenhypersurfuces,, \emph{Current Topics Inpurtiul Differential Equations}, (1986), 1.   Google Scholar

[5]

L. Caffarelli, L. Nirenberg and J. Spruck, Nonlinear second order elliptic equations V. The Dirichlet problem for Weingurten hypersurfuces,, \emph{Communications on Pure and Applied Mathematics}, 41 (1988), 47.  doi: 10.1002/cpa.3160410105.  Google Scholar

[6]

L. E. Fraenkel, An introduction to Maximum Principles and Symmetry in Elliptic Problems,, Cambridge Tracts in Mathematics, (2000).  doi: 10.1017/CBO9780511569203.  Google Scholar

[7]

B. Gidas, Weiming Ni and L. Nirenberg, Symmetry and related properties via the maximum principle,, \emph{Comm. Math. Phys.}, 68 (1979), 209.   Google Scholar

[8]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Second edition, (1983).  doi: 10.1007/978-3-642-61798-0.  Google Scholar

[9]

B. Kawohl, Symmetrzationor how to prove symmetry of solutions to a PDE,, \emph{Partial differential equations} (Praha), (1998), 214.   Google Scholar

[10]

Congming Li, Monotonicity and symmetry of solutions of fully nonlinear elliptic equations on unbounded domains,, \emph{Comm. Partial Differential Equations}, 16 (1991), 585.  doi: 10.1080/03605309108820770.  Google Scholar

[11]

W. Reichel, Radial symmetry for elliptic boundary-value problems on exterior domains,, \emph{Arch. Rational Mech. Anal.}, 137 (1997), 381.  doi: 10.1007/s002050050034.  Google Scholar

[12]

W. Reichel, Radial symmetry for an electrostatic, a capillarity and some fully nonlinear overdetermined problems on exterior domains,, \emph{Z. Anal. Anwendungen, 15 (1996), 619.  doi: 10.4171/ZAA/719.  Google Scholar

[13]

J. Serrin, A symmetry problem in potential theory,, \emph{Arch. Rational Mech. Anal.}, 43 (1971), 304.   Google Scholar

[14]

H. Shahgholian, Diversifications of Serrin's and related symmetry problems,, \emph{Complex Variables and Elliptic Equations, 57 (2012), 653.  doi: 10.1080/17476933.2010.504848.  Google Scholar

[15]

I. S. Sokolnikoff, Mathematical Theory of Elasticity,, McGraw-Hill, (1956).   Google Scholar

[16]

H. F. Weinberger, Remark on the preceding paper of Serrin,, \emph{Arch. Rational Mech. Anal.}, 43 (1971), 319.   Google Scholar

[1]

Huyuan Chen, Dong Ye, Feng Zhou. On gaussian curvature equation in $ \mathbb{R}^2 $ with prescribed nonpositive curvature. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3201-3214. doi: 10.3934/dcds.2020125

[2]

Anh Tuan Duong, Phuong Le, Nhu Thang Nguyen. Symmetry and nonexistence results for a fractional Choquard equation with weights. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 489-505. doi: 10.3934/dcds.2020265

[3]

Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $ p $-Laplacian. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020445

[4]

Yuxia Guo, Shaolong Peng. A direct method of moving planes for fully nonlinear nonlocal operators and applications. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020462

[5]

Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384

[6]

Vo Van Au, Hossein Jafari, Zakia Hammouch, Nguyen Huy Tuan. On a final value problem for a nonlinear fractional pseudo-parabolic equation. Electronic Research Archive, 2021, 29 (1) : 1709-1734. doi: 10.3934/era.2020088

[7]

Maho Endo, Yuki Kaneko, Yoshio Yamada. Free boundary problem for a reaction-diffusion equation with positive bistable nonlinearity. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3375-3394. doi: 10.3934/dcds.2020033

[8]

Shumin Li, Masahiro Yamamoto, Bernadette Miara. A Carleman estimate for the linear shallow shell equation and an inverse source problem. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 367-380. doi: 10.3934/dcds.2009.23.367

[9]

Helin Guo, Huan-Song Zhou. Properties of the minimizers for a constrained minimization problem arising in Kirchhoff equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1023-1050. doi: 10.3934/dcds.2020308

[10]

Stanislav Nikolaevich Antontsev, Serik Ersultanovich Aitzhanov, Guzel Rashitkhuzhakyzy Ashurova. An inverse problem for the pseudo-parabolic equation with p-Laplacian. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021005

[11]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319

[12]

Abdollah Borhanifar, Maria Alessandra Ragusa, Sohrab Valizadeh. High-order numerical method for two-dimensional Riesz space fractional advection-dispersion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020355

[13]

Maika Goto, Kazunori Kuwana, Yasuhide Uegata, Shigetoshi Yazaki. A method how to determine parameters arising in a smoldering evolution equation by image segmentation for experiment's movies. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 881-891. doi: 10.3934/dcdss.2020233

[14]

Yi Zhou, Jianli Liu. The initial-boundary value problem on a strip for the equation of time-like extremal surfaces. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 381-397. doi: 10.3934/dcds.2009.23.381

[15]

Yasmine Cherfaoui, Mustapha Moulaï. Biobjective optimization over the efficient set of multiobjective integer programming problem. Journal of Industrial & Management Optimization, 2021, 17 (1) : 117-131. doi: 10.3934/jimo.2019102

[16]

Marc Homs-Dones. A generalization of the Babbage functional equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 899-919. doi: 10.3934/dcds.2020303

[17]

Chang-Yeol Jung, Roger Temam. Interaction of boundary layers and corner singularities. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 315-339. doi: 10.3934/dcds.2009.23.315

[18]

Ying Liu, Yanping Chen, Yunqing Huang, Yang Wang. Two-grid method for semiconductor device problem by mixed finite element method and characteristics finite element method. Electronic Research Archive, 2021, 29 (1) : 1859-1880. doi: 10.3934/era.2020095

[19]

Julian Tugaut. Captivity of the solution to the granular media equation. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021002

[20]

Bilel Elbetch, Tounsia Benzekri, Daniel Massart, Tewfik Sari. The multi-patch logistic equation. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021025

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (24)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]