American Institute of Mathematical Sciences

Advanced
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, Arch. Rational Mech. Anal., 190 (2008), 267-280. 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, J. Differ. Equ., 245 (2008), 1566-1583. 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, Annali di Matematica., 188 (2009), 445-453. 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, Current Topics Inpurtiul Differential Equations, (1986), 1-26.  Google Scholar

[5]

L. Caffarelli, L. Nirenberg and J. Spruck, Nonlinear second order elliptic equations V. The Dirichlet problem for Weingurten hypersurfuces, Communications on Pure and Applied Mathematics, 41 (1988), 47-70. 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, 128. Cambridge University Press, Cambridge, 2000. doi: 10.1017/CBO9780511569203.  Google Scholar

[7]

B. Gidas, Weiming Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243.  Google Scholar

[8]

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

[9]

B. Kawohl, Symmetrzationor how to prove symmetry of solutions to a PDE, Partial differential equations (Praha), (1998), 214-229, Chapman & Hall/ CRC Res. Notes Math., 406, Chapman & Hall/ CRC, Boca Raton, FL, 2000.  Google Scholar

[10]

Congming Li, Monotonicity and symmetry of solutions of fully nonlinear elliptic equations on unbounded domains, Comm. Partial Differential Equations, 16 (1991), 585-615. doi: 10.1080/03605309108820770.  Google Scholar

[11]

W. Reichel, Radial symmetry for elliptic boundary-value problems on exterior domains, Arch. Rational Mech. Anal., 137 (1997), 381-394. 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, Z. Anal. Anwendungen, 15 (1996), 619-635. doi: 10.4171/ZAA/719.  Google Scholar

[13]

J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318.  Google Scholar

[14]

H. Shahgholian, Diversifications of Serrin's and related symmetry problems, Complex Variables and Elliptic Equations, 57 (2012), 653-665. doi: 10.1080/17476933.2010.504848.  Google Scholar

[15]

I. S. Sokolnikoff, Mathematical Theory of Elasticity, McGraw-Hill, New York, 1956.  Google Scholar

[16]

H. F. Weinberger, Remark on the preceding paper of Serrin, Arch. Rational Mech. Anal., 43 (1971), 319-320.  Google Scholar

show all references

References:
[1]

B. Brandolini, C. Nitsch, P. Salani and C. Trombetti, Serrin-type overdetermined problems: An alternative proof, Arch. Rational Mech. Anal., 190 (2008), 267-280. 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, J. Differ. Equ., 245 (2008), 1566-1583. 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, Annali di Matematica., 188 (2009), 445-453. 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, Current Topics Inpurtiul Differential Equations, (1986), 1-26.  Google Scholar

[5]

L. Caffarelli, L. Nirenberg and J. Spruck, Nonlinear second order elliptic equations V. The Dirichlet problem for Weingurten hypersurfuces, Communications on Pure and Applied Mathematics, 41 (1988), 47-70. 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, 128. Cambridge University Press, Cambridge, 2000. doi: 10.1017/CBO9780511569203.  Google Scholar

[7]

B. Gidas, Weiming Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243.  Google Scholar

[8]

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

[9]

B. Kawohl, Symmetrzationor how to prove symmetry of solutions to a PDE, Partial differential equations (Praha), (1998), 214-229, Chapman & Hall/ CRC Res. Notes Math., 406, Chapman & Hall/ CRC, Boca Raton, FL, 2000.  Google Scholar

[10]

Congming Li, Monotonicity and symmetry of solutions of fully nonlinear elliptic equations on unbounded domains, Comm. Partial Differential Equations, 16 (1991), 585-615. doi: 10.1080/03605309108820770.  Google Scholar

[11]

W. Reichel, Radial symmetry for elliptic boundary-value problems on exterior domains, Arch. Rational Mech. Anal., 137 (1997), 381-394. 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, Z. Anal. Anwendungen, 15 (1996), 619-635. doi: 10.4171/ZAA/719.  Google Scholar

[13]

J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318.  Google Scholar

[14]

H. Shahgholian, Diversifications of Serrin's and related symmetry problems, Complex Variables and Elliptic Equations, 57 (2012), 653-665. doi: 10.1080/17476933.2010.504848.  Google Scholar

[15]

I. S. Sokolnikoff, Mathematical Theory of Elasticity, McGraw-Hill, New York, 1956.  Google Scholar

[16]

H. F. Weinberger, Remark on the preceding paper of Serrin, Arch. Rational Mech. Anal., 43 (1971), 319-320.  Google Scholar

[1]

Roberto Triggiani. Unique continuation of boundary over-determined Stokes and Oseen eigenproblems. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 645-677. doi: 10.3934/dcdss.2009.2.645
[2]

Xinqun Mei, Jundong Zhou. The interior gradient estimate of prescribed Hessian quotient curvature equation in the hyperbolic space. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1187-1198. doi: 10.3934/cpaa.2021012
[3]

Bendong Lou. Traveling wave solutions of a generalized curvature flow equation in the plane. Conference Publications, 2007, 2007 (Special) : 687-693. doi: 10.3934/proc.2007.2007.687
[4]

Thomas Carty. Grossly determined solutions for a Boltzmann-like equation. Kinetic & Related Models, 2017, 10 (4) : 957-976. doi: 10.3934/krm.2017038
[5]

Hiroshi Matsuzawa. A free boundary problem for the Fisher-KPP equation with a given moving boundary. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1821-1852. doi: 10.3934/cpaa.2018087
[6]

Franco Obersnel, Pierpaolo Omari. On a result of C.V. Coffman and W.K. Ziemer about the prescribed mean curvature equation. Conference Publications, 2011, 2011 (Special) : 1138-1147. doi: 10.3934/proc.2011.2011.1138
[7]

Chiara Corsato, Franco Obersnel, Pierpaolo Omari, Sabrina Rivetti. On the lower and upper solution method for the prescribed mean curvature equation in Minkowski space. Conference Publications, 2013, 2013 (special) : 159-169. doi: 10.3934/proc.2013.2013.159
[8]

Hector D. Ceniceros. A semi-implicit moving mesh method for the focusing nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2002, 1 (1) : 1-18. doi: 10.3934/cpaa.2002.1.1
[9]

Tiancong Chen, Qing Han. Smooth local solutions to Weingarten equations and $\sigma_k$-equations. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 653-660. doi: 10.3934/dcds.2016.36.653
[10]

Shi Jin, Dongsheng Yin. Computational high frequency wave diffraction by a corner via the Liouville equation and geometric theory of diffraction. Kinetic & Related Models, 2011, 4 (1) : 295-316. doi: 10.3934/krm.2011.4.295
[11]

Shenghao Li, Min Chen, Bing-Yu Zhang. A non-homogeneous boundary value problem of the sixth order Boussinesq equation in a quarter plane. Discrete & Continuous Dynamical Systems, 2018, 38 (5) : 2505-2525. doi: 10.3934/dcds.2018104
[12]

Ka Luen Cheung, Man Chun Leung. Asymptotic behavior of positive solutions of the equation $ \Delta u + K u^{\frac{n+2}{n-2}} = 0$ in $IR^n$ and positive scalar curvature. Conference Publications, 2001, 2001 (Special) : 109-120. doi: 10.3934/proc.2001.2001.109
[13]

Simon Lloyd. On the Closing Lemma problem for the torus. Discrete & Continuous Dynamical Systems, 2009, 25 (3) : 951-962. doi: 10.3934/dcds.2009.25.951
[14]

D.J. Georgiev, A. J. Roberts, D. V. Strunin. Nonlinear dynamics on centre manifolds describing turbulent floods: k-$\omega$ model. Conference Publications, 2007, 2007 (Special) : 419-428. doi: 10.3934/proc.2007.2007.419
[15]

N. D. Alikakos, P. W. Bates, J. W. Cahn, P. C. Fife, G. Fusco, G. B. Tanoglu. Analysis of a corner layer problem in anisotropic interfaces. Discrete & Continuous Dynamical Systems - B, 2006, 6 (2) : 237-255. doi: 10.3934/dcdsb.2006.6.237
[16]

Alberto Bressan, Massimo Fonte. On the blow-up for a discrete Boltzmann equation in the plane. Discrete & Continuous Dynamical Systems, 2005, 13 (1) : 1-12. doi: 10.3934/dcds.2005.13.1
[17]

Huyuan Chen, Dong Ye, Feng Zhou. On gaussian curvature equation in $ \mathbb{R}^2 $ with prescribed nonpositive curvature. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 3201-3214. doi: 10.3934/dcds.2020125
[18]

Yu-Feng Sun, Zheng Zeng, Jie Song. Quasilinear iterative method for the boundary value problem of nonlinear fractional differential equation. Numerical Algebra, Control & Optimization, 2020, 10 (2) : 157-164. doi: 10.3934/naco.2019045
[19]

Tingzhi Cheng. Monotonicity and symmetry of solutions to fractional Laplacian equation. Discrete & Continuous Dynamical Systems, 2017, 37 (7) : 3587-3599. doi: 10.3934/dcds.2017154
[20]

Meng Qu, Ping Li, Liu Yang. Symmetry and monotonicity of solutions for the fully nonlinear nonlocal equation. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1337-1349. doi: 10.3934/cpaa.2020065

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (28)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences