Strain Tensors and Matching Property on Surfaces with the Gauss curvature changing sign

TL;DR

This paper proves regularity, density, and matching properties of strain tensors on surfaces with changing Gauss curvature, aiding the development of shell theories in elasticity.

Contribution

It establishes regularity, density of smooth isometries, and matching properties for strain tensors on surfaces with variable Gauss curvature, advancing shell theory analysis.

Findings

Proved regularity of solutions to strain tensor equations.

Established density of smooth infinitesimal isometries.

Demonstrated the matching property for surfaces with changing Gauss curvature.

Abstract

We prove the regularity of solutions to the strain tensor equation on a region $S$ with the Gauss curvature changing sign. Furthermore, we obtain the density property that smooth infinitesimal isometries are dense in the $W^{2,2}(S,\mathbb{R}^3)$ infinitesimal isometries. Finally, the matching property is established. Those results are important tools in obtaining recovery sequences ($\Gamma$-lim sup inequality) for dimensionally-reduced shell theories in elasticity.