Regularity of intrinsically convex $W^{2,2}$ surfaces and a derivation of a homogenized bending theory of convex shells

TL;DR

This paper proves smoothness of certain convex surfaces with $W^{2,2}$ regularity and derives a homogenized bending theory for convex shells with inhomogeneous energy, advancing understanding of shell mechanics.

Contribution

It establishes smoothness results for $W^{2,2}$ isometric immersions and derives a homogenized bending theory for convex shells with inhomogeneous energy density.

Findings

Proves smoothness of $W^{2,2}$ isometric immersions of convex surfaces.

Derives the $ ext{Gamma}$-limit for inhomogeneous shell energies in bending regime.

Provides a rigorous foundation for homogenized shell models.

Abstract

We prove smoothness of $W^{2,2}$ isometric immersions of surfaces endowed with a smooth Riemannian metric of positive Gauss curvature. We then derive the $\Gamma$-limit of three dimensional nonlinear shells with inhomogeneous energy density, in the bending energy regime.