# Integral Estimates for Approximations by Volume Preserving Maps

## TL;DR

This paper establishes quantitative bounds on how closely maps approximate volume preservation, linking deviations in derivatives to overall map deviations, with applications to elastic material deformations.

## Contribution

It introduces a Brenier decomposition-based method to quantify deviations from volume preservation and connects it to matrix nearness problems in elasticity theory.

## Key findings

- Bound on deviation of maps from volume preservation
- Relation between derivative deviation and map deviation
- Application to incompressible elastic deformations

## Abstract

A quantitative Brenier decomposition shows that the deviation of a map from volume preserving is bounded by the deviation of the derivative from volume preserving. A study of the matrix nearness problem for $SL(n)$ and $Sp(2n)$ relates the estimate to incompressible deformations of elastic materials.