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

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

Bo Wang 1, and  Jiguang Bao 1,

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.
Keywords: $k-$Hessian equation, moving plane method, Mirror symmetry, corner lemma., over-determined problem, Weingarten curvature equation.
Mathematics Subject Classification: Primary: 35J60; Secondary: 35J1.
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

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