# Morrey's conjecture: rank-one convexity implies quasi-convexity for two-dimensional, two-component maps

## TL;DR

This paper proves that for two-dimensional, two-component maps, rank-one convexity and quasiconvexity are equivalent, using a fixed-point argument to establish the connection and lamination representation.

## Contribution

It establishes the equivalence of rank-one convexity and quasiconvexity for two-component maps in two dimensions, a longstanding conjecture in the field.

## Key findings

- Rank-one convexity implies quasiconvexity for two-dimensional, two-component maps.
- A fixed-point argument ensures the existence of lamination representations.
- Higher-dimensional cases remain unresolved and require further geometric insights.

## Abstract

We prove that for two-component maps in dimension two, rank-one convexity is equivalent to quasiconvexity. The essential tool for the proof is a fixed-point argument for a suitable set-valued map going from one component to the other that preserves decomposition directions within the $(H_n)$-condition formalism. The existence of a fixed point ensures that, in addition to keeping decomposition directions, joint volume fractions are respected as well, leading to the fundamental fact that every two-dimensional, two-component gradient can be reached by lamination. When maps have more than two components, fixed points exist for every combination of two components, but they do not match in general. Higher dimension would require further insight on how to organize and deal with triangulations for piece-wise affine maps.