Gelfand problem and Hemisphere rigidity

TL;DR

This paper connects the hemisphere rigidity theorem to the Gelfand problem, providing new insights into conformal geometry and nonlinear PDEs, especially for bi-Laplacian equations with curvature bounds.

Contribution

It generalizes the hemisphere rigidity theorem using Q-curvature bounds and interprets it within the framework of the fourth order Gelfand problem for bi-Laplacian equations.

Findings

Established a link between hemisphere rigidity and Gelfand problem solutions.

Derived explicit extremal values for specific nonlinearities in 2D and higher dimensions.

Extended rigidity results to Q-curvature lower bounds and bi-Laplacian equations.

Abstract

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 $-\Delta u=\lambda g(u)$ for certain nonlinearity $g$ in a bounded region $\Omega \subset \mathbb{R}^n$ subject to the Dirichlet boundary condition. It is well-known that there exists an extremal $\lambda^{*}$, such that for $\lambda>\lambda^{*}$, the above equation does not admit any solution. Interestingly, Hang-Wang's hemisphere rigidity theorem yields a precise value for $\lambda^{*}$ 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.