Rigidity Estimate for Hyperbolic Shells and its Application in $\mathrm{\Gamma }$-limit Theory

This paper establishes novel rigidity estimates for hyperbolic shells (surfaces with negative Gaussian curvature) and applies them to derive the $\mathrm{\Gamma }$-limit of thin elastic shells. We prove a nonlinear rigidity estimate for $H^1$ deformations on the mid-surface, and a nonlinear rigidity estimate for hyperbolic shells with clamped lateral boundary. The latter yields the optimal exponent $h^{-4\mathrm{/}3}$. As the main application, we characterize the $\mathrm{\Gamma }$-limit of the nonlinear elastic energy for clamped hyperbolic shells across all scaling regimes $\beta \in [0,2)\cup (8\mathrm{/}3,\mathrm{\infty })$.