环形区域上Serrin问题的非平凡解
Nontrivial solutions to Serrin's problem in annular domains
摘要：
我们构造非平凡光滑有界区域$Ω\subseteq \mathbb{R}^n$的形式$Ω_0 \setminus \overlineΩ_1$，从annuli分叉，其中存在一个正解的超定边值问题\[ -Δu = 1, \; u>0 \quad \text{in } Ω, \qquad u = 0 ,\; \partial_νu = \text{const} \quad \text{on } \partialΩ_0, \qquad u = \text{const} ,\; \partial_νu = \text{const} \quad \text{on } \partial Ω_1, \]，其中$ν$代表内部单位正常$\partialΩ$。从Reichel和后来的Sirakov的结果中，人们知道$\partialΩ_1$上的条件$\partial_νu \leq 0$足以使刚性保持，即只有允许这样的解的域是环形的，并且解是径向对称的。我们的构造表明，这个条件也是必要的。此外，所构建的结构域被证明是自我Cheeger。
英文摘要：
We construct nontrivial smooth bounded domains $Ω\subseteq \mathbb{R}^n$ of the form $Ω_0 \setminus \overlineΩ_1$, bifurcating from annuli, for which there exists a positive solution to the overdetermined boundary value problem \[ -Δu = 1, \; u>0 \quad \text{in } Ω, \qquad u = 0 ,\; \partial_νu = \text{const} \quad \text{on } \partialΩ_0, \qquad u = \text{const} ,\; \partial_νu = \text{const} \quad \text{on } \partial Ω_1, \] where $ν$ stands for the inner unit normal to $\partialΩ$. From results by Reichel and later by Sirakov, it was known that the condition $\partial_νu \leq 0$ on $\partialΩ_1$ is sufficient for rigidity to hold, namely, the only domains which admit such a solution are annuli and solutions are radially symmetric. Our construction shows that the condition is also necessary. In addition, the constructed domains are shown to be self-Cheeger.
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