# A primal dual variational formulation and a multi-duality principle for a non-linear model of plates

## TL;DR

This paper introduces a new primal dual variational formulation and a multi-duality principle for a non-linear plate model, providing conditions for global optimality and analyzing duality gaps.

## Contribution

It develops a novel primal dual formulation and multi-duality principle for the Kirchhoff-Love non-linear plate model, including global optimality conditions.

## Key findings

- Established a duality principle suitable for negative definite membrane stress tensors.
- Developed a primal dual variational formulation with sufficient conditions for global optimality.
- Proved there is no duality gap between primal and dual formulations in a local extremal context.

## Abstract

This article develops a new primal dual formulation for the Kirchhoff-Love non-linear plate model. At first we establish a duality principle which includes sufficient conditions of global optimality through the dual formulation. At this point we highlight this first duality principle is specially suitable for the case in which the membrane stress tensor is negative definite. In a second step, from such a general principle, we develop a primal dual variational formulation which also includes the corresponding sufficient conditions for global optimality. The results are based on standard tools of convex analysis and on a well known Toland result for D.C. optimization. Finally, in the last section, we present a multi-duality principle and qualitative relations between the critical points of the primal and dual formulations. We formally prove there is no duality gap between such primal and dual formulations in a local extremal context.