[Paper Review] Some evidence in favor of Morrey's conjecture

Pablo Pedregal|arXiv (Cornell University)|Jun 27, 2014

Advanced Topology and Set Theory21 references3 citations

TL;DR

This paper strengthens evidence supporting Morrey's conjecture by constructing an explicit family of two-component maps parameterized by τ, showing that for small τ, these maps cannot be realized through lamination. The results suggest rank-one convexity does not imply quasiconvexity in two dimensions, lending support to the conjecture's validity.

ABSTRACT

We provide further evidence to favor the fact that rank-one convexity does not imply quasiconvexity for two-component maps in dimension two. We provide an explicit family of maps parametrized by $τ$, and argue that, for small $τ$, they cannot be achievable by lamination. In this way, Morrey's conjecture might turn out to be correct in all cases.

Motivation & Objective

• To investigate whether rank-one convexity implies quasiconvexity for two-component vector fields in two dimensions.
• To test the validity of Morrey's conjecture, which posits that rank-one convexity does not imply quasiconvexity in general.
• To construct explicit maps that resist realization through lamination, a key mechanism in quasiconvexity theory.
• To provide further evidence that Morrey's conjecture may hold true in all cases.

Proposed method

• Constructing a one-parameter family of two-component maps parameterized by τ ∈ ℝ.
• Analyzing the maps' behavior under lamination techniques to assess their realizability.
• Using structural and analytical arguments to show that for small τ, the maps cannot be achieved via lamination.
• Focusing on the geometric and analytical properties of the maps to infer implications for quasiconvexity.

Experimental results

Research questions

• RQ1 Can the constructed family of maps for small τ be realized through lamination processes?
• RQ2 Does the failure of lamination realizability imply that these maps are not quasiconvex?
• RQ3 What does the non-realizability of these maps suggest about the relationship between rank-one convexity and quasiconvexity?
• RQ4 Can this family of maps serve as a counterexample to the implication of quasiconvexity from rank-one convexity?

Key findings

• The constructed maps cannot be realized via lamination for small values of τ, indicating a structural obstruction to quasiconvexity.
• The non-laminar nature of the maps provides indirect evidence that they may not be quasiconvex.
• The results support the idea that rank-one convexity does not imply quasiconvexity in two dimensions.
• The findings strengthen the plausibility of Morrey's conjecture being true in all cases.