This paper considers overdetermined boundary problems. Firstly, the author gives a proof of the Payne-Schaefer conjecture about an overdetermined problem of sixth order in the two-dimensional case and under an additional condition for the case of dimension no less than three. Secondly, the author proves an integral identity for an overdetermined problem of fourth order which can be used to deduce Bennett’s symmetry theorem. Finally, the author proves a symmetry result for an overdetermined problem of second order by integral identities.
| [1] |
Alessandrini G. A symmetry theorem for condensers. Math. Methods Appl. Sci., 1992, 15: 315-320
|
| [2] |
Amdeberhan, T., Two symmetry problems in potential theory, Electron. J. Differential Equations, 2001, Paper No. 43, 5pp.
|
| [3] |
Bennett A. Symmetry in an overdetermined fourth order elliptic boundary value problem. SIAM J. Math. Anal., 1986, 17: 1354-1358
|
| [4] |
Brandolini B, Nitsch C, Salani P, Trombetti C. Serrin type overdetermined problems: An alternative proof. Arch. Rat. Mech. Anal., 2008, 190: 267-280
|
| [5] |
Brandolini B, Nitsch C, Salani P, Trombetti C. On the stability of the Serrin problem. J. Differential Equations, 2008, 245: 1566-1583
|
| [6] |
Choulli M, Henrot A. Use of the domain derivative to prove symmetry results in partial differential equations. Math. Nachr., 1998, 192: 91-103
|
| [7] |
Farina A, Kawohl B. Remarks on an overdetermined boundary value problem. Calc. Var. Partial Differential Equations, 2008, 31(3): 351-357
|
| [8] |
Garofalo N, Lewis J. A symmetry result related to some overdetermined boundary value problems. Amer. J. Math., 1989, 111: 9-33
|
| [9] |
Payne L E. Some remarks on maximum principles. J. Analyse Math., 1976, 30: 421-433
|
| [10] |
Payne L E, Schaefer P W. Duality theorems in some overdetermined problems. Math. Methods Appl. Sci., 1989, 11: 805-819
|
| [11] |
Philippin G A, Ragoub L. On some second order and fourth order elliptic overdetermined problems. Z. Angew. Math. Phys., 1995, 46: 188-197
|
| [12] |
Serrin J. A symmetry problem in potential theory. Arch. Rational Mech. Anal., 1971, 43: 304-318
|
| [13] |
Troy W C. Symmetry properties in systems of semilinear elliptic equations. J. Differential Equations, 1981, 42: 400-413
|
| [14] |
Weinberger H. Remark on the preceding paper of Serrin. Arch. Rational Mech. Anal., 1971, 43: 319-320
|
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