APM  Vol.1 No.6 , November 2011
Some Steiner Symmetry Results in Overdetermined Boundary Value Problem
ABSTRACT
In this paper, we use the moving planes method to prove that the domain Ω and the solution u are Steiner symmetric if u is a positive solution of the overdetermined boundary value problem in Ω.

Cite this paper
nullZ. Fang and A. Wang, "Some Steiner Symmetry Results in Overdetermined Boundary Value Problem," Advances in Pure Mathematics, Vol. 1 No. 6, 2011, pp. 340-344. doi: 10.4236/apm.2011.16061.
References
[1]   J. Serrin, “A Symmetry Problem in Potential Theory,” Archive for Rational Mechanics and Analysis, Vol. 43, No. 4, 1971, pp. 304-318. doi:10.1007/BF00250468

[2]   B. Gidas, W. M. Ni and L. Nirenberg, “Symmetry and Related Properties via the Maximum Principle,” Communications in Mathematical Physics, Vol. 68, No. 3, 1979, pp. 209-243. doi:10.1007/BF01221125

[3]   C. H. Kim, “Steiner Symmetry in Overdetermined Boundary Value Problems,” Doctor’s Degree Thesis, Chonnam National University, Kwangju, Korea, 2002.

[4]   A. Colesanti, “A Symmetry Result for the p-Laplacian Equation via the Moving Planes Method,” Applicable Analysis, Vol. 55, No. 3-4, 1994, pp. 207-213.

[5]   F. Brock, A. Henrot, “A Symmetry Result for an Overdetermined Elliptic Problem Using Continuous Rearrangement and Domain Derivative,” Rendiconti del Circolo Matematico di Palermo, Vol. 51, No. 3, 2002, pp. 375-390. doi:10.1007/BF02871848

[6]   M. Choulli and A. Henrot, “Use of the Domain Derivative to Prove Symmetry Results in Partial Differential Equations,” Mathematische Nachrichten, Vol. 192, 1998, pp. 91-103. doi:10.1002/mana.19981920106

[7]   I. Fragala, I. F. Gazzaola and B. Kawohl, “Overdetemined Boundary Value Problems with Possibly Degenerate Ellipticity: A Geometry Approach,” Mathematische Zeitschrift, Vol. 254, No. 1, 2006, pp. 117-132. doi:10.1007/s00209-006-0937-7

[8]   N. Garofalo and J. Lewis, “A Symmetry Result Related to Some Overdeternined Boundary Value Problems,” American Journal of Mathematics, Vol. 111, No. 1, 1989, pp. 9-33. doi:10.2307/2374477

[9]   L. Ragous, “Symmetry Theorems via the Continuous Steiner Symmetrization,” Electronic Journal of Differential Equations, Vol. 2000, No. 44, 2000, pp. 1-11.

 
 
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