2021/02/20 by Mijia Lai, Wei Wei, Lai, Mijia +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2102.10360
openalex publication_date 2021/02/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 g0 on Sn+ with R≥ n(n-1) and whose boundary coincides with g0|_∂ Sn+, then g=g0. This is related to the classical Gelfand problem, which investigates -Δu=λg(u) for certain nonlinearity g in a bounded region Ω⊂ ℝn 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)=e2u when n=2 and g(u)=(1+u)(n+2)/(n-2) for n≥ 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.