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Some Steiner Symmetry Results in Overdetermined Boundary Value Problem
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Advances in Pure Mathematics·Volume 01 (2011)·Pages 340–344·Published 24 November 2011·DOI10.4236/apm.2011.16061
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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.
KeywordsSteiner SymmetryMoving Planes MethodOverdetermined Boundary Value Problem
- 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
- 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
- C. H. Kim, “Steiner Symmetry in Overdetermined Boundary Value Problems,” Doctor’s Degree Thesis, Chonnam National University, Kwangju, Korea, 2002.
- 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.
- 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
- 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
- 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
- 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
- L. Ragous, “Symmetry Theorems via the Continuous Steiner Symmetrization,” Electronic Journal of Differential Equations, Vol. 2000, No. 44, 2000, pp. 1-11.