Rigidity Estimate for Hyperbolic Shells and its Application in $Γ$-limit Theory

Abstract

This paper establishes novel rigidity estimates for hyperbolic shells (surfaces with negative Gaussian curvature) and applies them to derive the \(Γ\)-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 \(h^{-4/3}\). As the main application, we characterize the \(Γ\)-limit of the nonlinear elastic energy for clamped hyperbolic shells across all scaling regimes \(β\in [0,2) \cup (8/3,\infty)\).