Gelfand问题与半球刚性
Gelfand problem and Hemisphere rigidity
摘要：
本文在Gelfand问题的框架下给出了Hang-Wang的半球刚性定理的一个解释。更精确地说，Hang-Wang证明了，对于一个度量$g$，它与$S^{n}_{+}$上的标准度量$g_0$共形，且其边界与$g_0|_{\partial S^{n}_{+}}$重合，则$g=g_0$。这与经典的Gelfand问题有关，该问题研究了在Dirichlet边界条件下有界区域$Ω\subset \mathbb{R}^n$中某些非线性$g$的$-Δu=λg(u)$。众所周知，存在极值$λ^{*}$，使得对于$λ>λ^{*}$，上述方程不允许任何解。有趣的是，当$n=2$和$g(u)=(1+u)^{\frac{n+2}{n-2}}$对于$n\geq 3$时，Hang-Wang的半球刚性定理对于$g(u)=e^{2u}$给出了$λ^{*}$的精确值。我们试图推广的半球刚性定理下$Q$曲率下界和适合这到四阶Gelfand问题的双拉普拉斯共形非线性的解释。
英文摘要：
We give an interpretation of the hemisphere rigidity theorem of Hang-Wang in the framework of Gelfand problem. More precisely, Hang-Wang showed that for a metric $g$ conformal to the standard metric $g_0$ on $S^{n}_{+}$ with $R\geq n(n-1)$ and whose boundary coincides with $g_0|_{\partial S^{n}_{+}}$, then $g=g_0$. This is related to the classical Gelfand problem, which investigates $-Δu=λg(u)$ for certain nonlinearity $g$ in a bounded region $Ω\subset \mathbb{R}^n$ subject to the Dirichlet boundary condition. It is well-known that there exists an extremal $λ^{*}$, such that for $λ>λ^{*}$, the above equation does not admit any solution. Interestingly, Hang-Wang's hemisphere rigidity theorem yields a precise value for $λ^{*}$ for $g(u)=e^{2u}$ when $n=2$ and $g(u)=(1+u)^{\frac{n+2}{n-2}}$ for $n\geq 3$. We attempt to generalize the hemisphere rigidity theorem under $Q$ curvature lower bound and fit this into the interpretation of fourth order Gelfand problem for bi-Laplacian with conformal nonlinearity.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
