# American Institute of Mathematical Sciences

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 and 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. [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. [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. [4] L. Caffarelli, L. Nirenberg and J. Spruck, Nonlinear second order equations IV. Starshaped compact Weingurtenhypersurfuces, Current Topics Inpurtiul Differential Equations, (1986), 1-26. [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. [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. [7] B. Gidas, Weiming Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243. [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. [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. [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. [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. [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. [13] J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318. [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. [15] I. S. Sokolnikoff, Mathematical Theory of Elasticity, McGraw-Hill, New York, 1956. [16] H. F. Weinberger, Remark on the preceding paper of Serrin, Arch. Rational Mech. Anal., 43 (1971), 319-320.

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##### 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. [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. [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. [4] L. Caffarelli, L. Nirenberg and J. Spruck, Nonlinear second order equations IV. Starshaped compact Weingurtenhypersurfuces, Current Topics Inpurtiul Differential Equations, (1986), 1-26. [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. [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. [7] B. Gidas, Weiming Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243. [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. [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. [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. [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. [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. [13] J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318. [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. [15] I. S. Sokolnikoff, Mathematical Theory of Elasticity, McGraw-Hill, New York, 1956. [16] H. F. Weinberger, Remark on the preceding paper of Serrin, Arch. Rational Mech. Anal., 43 (1971), 319-320.
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