Symmetric Solutions of a Nonlinear Elliptic Problem with Neumann Boundary Condition

Affiliation(s)

Group of Waves, Program of Mathematics, Faculty of Exact and Natural Sciences, Cartagena University, Cartagena de Indias, Co-lombia.

Morelia Institute of Technology, Morelia, Mexico.

Group of Waves, Program of Mathematics, Faculty of Exact and Natural Sciences, Cartagena University, Cartagena de Indias, Co-lombia.

Morelia Institute of Technology, Morelia, Mexico.

ABSTRACT

We show a result of symmetry for a big class of problems with condition of Neumann on the boundary in the case one dimensional. We use the method of reflection of Alexandrov and we show one application of this method and the maximum principle for elliptic operators in problems with conditions of Neumann. Some results of symmetry for elliptic problems with condition of Neumann on the boundary may be extended to elliptic operators more general than the Laplacian.

We show a result of symmetry for a big class of problems with condition of Neumann on the boundary in the case one dimensional. We use the method of reflection of Alexandrov and we show one application of this method and the maximum principle for elliptic operators in problems with conditions of Neumann. Some results of symmetry for elliptic problems with condition of Neumann on the boundary may be extended to elliptic operators more general than the Laplacian.

Cite this paper

A. Ramirez, R. Ortiz and J. Ceballos, "Symmetric Solutions of a Nonlinear Elliptic Problem with Neumann Boundary Condition,"*Applied Mathematics*, Vol. 3 No. 11, 2012, pp. 1686-1688. doi: 10.4236/am.2012.311233.

A. Ramirez, R. Ortiz and J. Ceballos, "Symmetric Solutions of a Nonlinear Elliptic Problem with Neumann Boundary Condition,"

References

[1] A. D. Alexandrov, “Uniqueness Theorems for Surfaces in the Large,” Vestnik Leningrad University: Mathematics, Vol. 13, No. 19, 1958, pp. 5-8.

[2] J. Serrin, “A Symetry Problem in Potential Theory,” Archive for Rational Mechanics and Analysis, Vol. 43, No. 4, 1971, pp. 304-318. doi:10.1007/BF00250468

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

[4] B. Gidas, W.-M. Ni and L. Nirenberg, “Symmetry of Positive Solutions of Nonlinear Elliptic Equations in ,” In: Mathematical Analysis and Applications, Part A, Academic Press, New York, 1981, pp. 369-402.

[5] L. Cafarelli, B. Gidas and J. Spruck, “Asymptotic Symmetry and Local Behavior of Semilinear Elliptic with Critical Sobolev Growth,” Communications on Pure and Applied Mathematics, Vol. 42, No. 3, 1989, pp. 271-297. doi:10.1002/cpa.3160420304

[6] H. Berestycki and L. Nirenberg, “On the Method of Moving Planes and the Sliding Method,” Bulletin of the Brazilian Mathematical Society, Vol. 22, No. 1, 1991, pp. 1-37.

[7] F. John, “Partial Differential Equations,” Springer-Verlag, New York, 1982.

[8] M. Protter and H. Weinberger, “Maximum Principle in Differential Equations,” Springer-Verlag, New York, 1984. doi:10.1007/978-1-4612-5282-5

[9] D. Gilbarg and N. Trudinger, “Elliptic Partial Differential Equations of Second Order,” Springer-Verlag, Berlin, Heidelberg, New York, 1977.

[1] A. D. Alexandrov, “Uniqueness Theorems for Surfaces in the Large,” Vestnik Leningrad University: Mathematics, Vol. 13, No. 19, 1958, pp. 5-8.

[2] J. Serrin, “A Symetry Problem in Potential Theory,” Archive for Rational Mechanics and Analysis, Vol. 43, No. 4, 1971, pp. 304-318. doi:10.1007/BF00250468

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

[4] B. Gidas, W.-M. Ni and L. Nirenberg, “Symmetry of Positive Solutions of Nonlinear Elliptic Equations in ,” In: Mathematical Analysis and Applications, Part A, Academic Press, New York, 1981, pp. 369-402.

[5] L. Cafarelli, B. Gidas and J. Spruck, “Asymptotic Symmetry and Local Behavior of Semilinear Elliptic with Critical Sobolev Growth,” Communications on Pure and Applied Mathematics, Vol. 42, No. 3, 1989, pp. 271-297. doi:10.1002/cpa.3160420304

[6] H. Berestycki and L. Nirenberg, “On the Method of Moving Planes and the Sliding Method,” Bulletin of the Brazilian Mathematical Society, Vol. 22, No. 1, 1991, pp. 1-37.

[7] F. John, “Partial Differential Equations,” Springer-Verlag, New York, 1982.

[8] M. Protter and H. Weinberger, “Maximum Principle in Differential Equations,” Springer-Verlag, New York, 1984. doi:10.1007/978-1-4612-5282-5

[9] D. Gilbarg and N. Trudinger, “Elliptic Partial Differential Equations of Second Order,” Springer-Verlag, Berlin, Heidelberg, New York, 1977.