AI 中文总结
该研究针对$\boldsymbol{\text{R}}^{2\times m}$中的莫雷问题，构造出当$m$足够大时处处非拟凸的齐次秩一凸被积函数，解决了相关问题
AI 中文摘要
当$m$足够大时，我们构造了从$\boldsymbol{\text{R}}^{2\times m}$到$\boldsymbol{\text{R}}$的齐次秩一凸被积函数$F$，其处处非拟凸
英文摘要
We construct, for any exponent $p\in(1,\infty)$, $p$-homogeneous rank-one convex integrands $F\colon \mathbb{R}^{2\times m}\to \mathbb{R}$ that are nowhere quasiconvex when $m$ is large. When $p$ is sufficiently close to $4$, such examples can be constructed on $\mathbb{R}^{2\times 3}$: for $p=4$, our example is an explicit quartic polynomial. Related constructions give, for every $p$, conjugation- and transposition-invariant examples on $\mathbb{R}^{d\times d}$, and examples on $\mathbb{R}^{4\times 2}$ for every $p\neq 2$.
Comments30 pages. The new version includes new examples in dimension 2x3 and 3x2, and some proofs have been streamlined. Comments are more than welcome
