Rigidity of Strain Tensors for Surfaces With Mixed Type and Applications To Shell Theory

Abstract

This paper investigates the rigidity of strain tensors on surfaces with sign-changing Gaussian curvature (mixed-type surfaces) and applies the results to determine the optimal thickness exponent in the first Korn inequality for thin shells. Using tools from Riemannian geometry and generalized tensor analysis, we derive an infinitesimal rigidity lemma for strain tensors, which establishes \(L^2\) regularity estimates for displacements decomposed into tangential and normal components. Specifically, we show that for a mixed-type shell with a middle surface \(S = S^+ \cup \Gamma_0 \cup S^-\) (where \(S^+\), \(S^-\) have positive and negative curvature, respectively, and \(\Gamma_0\) is a parabolic interface), the optimal constant in Korn's inequality scales as \(h^{4/3}\), matching the behavior previously established for hyperbolic shells. This result is obtained via a combination of geometric decomposition, Fredholm theory for linear operators, and compactness arguments to handle the curvature transition across \(\Gamma_0\). The findings bridge the gap between elliptic and hyperbolic shell theories, providing a unified framework for understanding rigidity in complex geometries with mixed curvature. The derived estimates are shown to be sharp, offering critical insights for the mechanical design of thin-walled structures with non-uniform curvature.