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arXiv · 2403.20265

Rigidity of Euclidean product structure: breakdown for low Sobolev exponents

Bruce Kleiner·Stefan Müller·László Székelyhidi Jr.·Xiangdong Xie
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Abstract

We develop a general toolbox to study $W^{1,p}$ solutions of differential inclusions $\nabla u \in K$ for unbounded sets $K$. A key notion is the concept that a subset $K$ of the space $\mathbb{R}^{d \times m}$ of $d \times m$ matrices can be reduced to another set $K'$. We then use this framework to show that the product rigidity for Sobolev maps fails for $p<2$, and also apply our toolbox to simplify several examples from the literature.

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Bruce Kleiner, Stefan Müller, László Székelyhidi Jr., Xiangdong Xie. 2024-03-29. Rigidity of Euclidean product structure: breakdown for low Sobolev exponents. https://arxiv.org/abs/2403.20265

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