Strain Tensors and Matching Property on Degenerated Hyperbolic Surfaces

TL;DR

This paper proves regularity, density, and matching properties of solutions to strain tensor equations on degenerated hyperbolic surfaces, aiding in the development of shell theories in elasticity.

Contribution

It establishes new regularity, density, and matching results for strain tensors on degenerated hyperbolic surfaces, crucial for elasticity shell models.

Findings

Proved regularity of solutions on degenerated hyperbolic surfaces.

Established density of smooth infinitesimal isometries in $W^{2,2}$.

Proved the matching property for these surfaces.

Abstract

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