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.
From: Pablo Pedregal [view email]
[v1]
Thu, 16 May 2019 07:30:17 UTC (20 KB)
[v2]
Fri, 27 Sep 2019 07:53:39 UTC (21 KB)
[v3]
Thu, 29 Aug 2024 08:25:22 UTC (1 KB) (withdrawn)
[v4]
Mon, 12 May 2025 18:04:47 UTC (557 KB)
[v5]
Thu, 18 Jun 2026 07:34:31 UTC (198 KB)
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