论文标题
在有限域中过度确定的椭圆问题的非对称签名解决方案
Nonsymmetric sign-changing solutions to overdetermined elliptic problems in bounded domains
论文作者
论文摘要
1971年，J。Serrin证明，给定一个平滑的有限域$ω\ subset \ mathbb {r}^n $和$ u $和$ u $的积极解决方案： \ begin {equation*} \ begin {array} {ll} -ΔU= f（u）＆\ mbox {in $ω$，} u = 0＆\ mbox {on $ \ \partialΩ$，} \partial_νu = \ mbox {constant}＆\ mbox {on $ \partialΩ$，} \ end {array} \ end {equation*}，然后$ω$必须是一个球，$ u $是径向对称的。在本文中，我们证明$ u $的阳性在对称结果中是必要的。实际上，我们在一个有限的域中$ c^2 $ function $ f（u）$ a $ω$在与球中不同的签名解决方案。根据相关线性化操作员的研究，该证明使用局部分叉参数。
In 1971 J. Serrin proved that, given a smooth bounded domain $Ω\subset \mathbb{R}^N$ and $u$ a positive solution of the problem: \begin{equation*} \begin{array}{ll} -Δu = f(u) &\mbox{in $Ω$, } u =0 &\mbox{on $\partialΩ$, } \partial_ν u =\mbox{constant} &\mbox{on $\partialΩ$, } \end{array} \end{equation*} then $Ω$ is necessarily a ball and $u$ is radially symmetric. In this paper we prove that the positivity of $u$ is necessary in that symmetry result. In fact we find a sign-changing solution to that problem for a $C^2$ function $f(u)$ in a bounded domain $Ω$ different from a ball. The proof uses a local bifurcation argument, based on the study of the associated linearized operator.
