带有非常数边界条件的各向异性超定问题

数学物理学报 ›› 2016, Vol. 36 ›› Issue (1): 27-35.

带有非常数边界条件的各向异性超定问题

阮其华

莆田学院数学学院福建莆田 351100

收稿日期:2015-08-04 修回日期:2015-12-17 出版日期:2016-02-25 发布日期:2016-02-25
作者简介:阮其华,ruanqihua@163.com
基金资助:
国家自然科学基金(11471175)和福建省自然科学基金(2012J01015)资助

An Overdetermined Anisotropic Problem with Nonconstant Boundary Condition

Ruan Qihua

School of Mathematics, Putian University, Fujian Putian 351100

Received:2015-08-04 Revised:2015-12-17 Online:2016-02-25 Published:2016-02-25
Supported by:
Supported by the NSFC (11471175) and the NSFC of Fujian Province (2012J01015)

摘要/Abstract

摘要：

Rⁿ中给定一开的有界连通子集Ω和范数H(ξ),考虑各向异性超定问题-div(H(∇u)∇_ξH(∇u))=1,在边界∂Ω上满足非常数边界条件:H(∇u)²=cx·∇u和u=0.如果这个各向异性超定问题有弱解,那么Ω具有Wulff形状.

关键词: 超定问题, 各向异性椭圆方程, Wulff形状

Abstract:

Given an open bounded connected subset Ω of Rⁿ with a norm H(ξ), we consider an overdetermined anisotropic problem of -div(H(∇u)∇_ξH(∇u))=1 with the the following nonconstant boundary condition: H(∇u)²=cx·∇u and u=0 on the boundary ∂Ω. If this overdetermined anisotropic problem admits a solution in a suitable weak sense, then Ω is of the Wulff Shape.

Key words: Overdetermined problem, Anisotropic elliptic equation, Wulff shape

中图分类号:

O186.1

阮其华. 带有非常数边界条件的各向异性超定问题[J]. 数学物理学报, 2016, 36(1): 27-35.

Ruan Qihua. An Overdetermined Anisotropic Problem with Nonconstant Boundary Condition[J]. Acta mathematica scientia,Series A, 2016, 36(1): 27-35.

参考文献

[1] Amdeberhan T. Two symmetry problems in potential theory. Electron J Differential Equations, 2001, 43:1-5
[2] Bianchini C, Henrot A, Salani P. An overdetermined problem with non-constant boundary condition. Interfaces Free Bound, 2014, 16(2):215-241
[3] Brandolini B, Nitsch C, Salani P, Trombetti C. Serrin-type overdetermined problems:an alternative proof. Arch Ration Mech Anal, 2008, 190:267-280
[4] Choulli M, Henrot A. Use of the domain derivative to prove symmetry results in partial differential equations. Math Nachr, 1998, 192:91-103
[5] Cianchi A, Salani P. Overdetermined anisotropic elliptic problems. Math Ann, 2009, 345:859-881
[6] Fragalá L. Symmetry results for overdetermined problems on convex domains via Brunn-Minkowski inequalities. J Math Pures Appl, 2012, 97(1):55-65
[7] Lieb E H, Loss M. Analysis. New Delhi:Narosa Publishing House, 1997
[8] Lv B, Li F, Zou W. Overdetermined boundary value problems with strongly nonlinear elliptic PDE. Electronic Journal of Qualitative Theory of Differential Equations, 2012, 10:1-17
[9] Greco A. Radial symmetry and uniqueness for an overdetermined problem. Math Methods Appl Sci, 2001, 24:103-115
[10] Henrot A, Philippin G. Some overdetermined boundary value problems with elliptical free boundaries. SIAM J Math Anal, 1998, 29:309-320
[11] Molzon R. Symmetry and overdetermined boundary value problems. Forum Math, 1991, 3(2):143-156
[12] Payne L E, Schaefer P W. On overdetermined boundary value problems for the biharmonic operator. Journal of Mathematical Analysis and Applications, 1994, 187(2):598-616
[13] Serrin J. A symmetry problem in potential theory. Arch Ration Mech Anal, 1971, 43:304-318
[14] Weinberger H. Remark on the preceding paper of Serrin. Arch Ration Mech Anal, 1971, 43:319-320
[15] Wang G, Xia C. A characterization of the Wulff shape by an overdetermined anisotropic PDE. Arch Ration Mech Anal, 2011, 199(1):99-115
[16] Wang L, Wang T. A note on an overdetermined system involving mean curvature. J Math Anal Appl, 2012, 393(2):489-492