# Polyconvexity and Existence Theorem for Nonlinearly Elastic Shells

## TL;DR

This paper proves an existence theorem for nonlinearly elastic shells using polyconvexity conditions, ensuring the existence of global energy minimizers in a low-regularity, two-dimensional shell model.

## Contribution

It introduces a new existence theorem for elastic shells under polyconvexity, expanding the mathematical understanding of shell stability with low regularity assumptions.

## Key findings

- Existence of global minimizers for elastic shells under specific conditions.
- Definition of a broad class of polyconvex stored energy functions.
- Establishment of coerciveness and growth conditions for energy functions.

## Abstract

We present an existence theorem for a large class of nonlinearly elastic shells with low regularity in the framework of a two-dimensional theory involving the mean and Gaussian curvatures. We restrict our discussion to hyperelastic materials, that is to elastic materials possessing a stored energy function. Under some specific conditions of polyconvexity, coerciveness and growth of the stored energy function, we prove the existence of global minimizers. In addition, we define a general class of polyconvex stored energy functions which satisfies a coerciveness inequality.