关于具有 $L^p$ 熵产生量的 Aviles-Giga 极限状态
On Aviles-Giga limit states with $L^p$ entropy productions
AI总结：
本文研究了 Aviles-Giga 能量下收敛到 eikonal 方程弱解的映射序列，探讨了熵产生量的分布特性，证明在特定条件下熵产生量可由 $L^p$ 函数表示。
摘要：
Aviles-Giga 能量提供了收敛到 eikonal 方程弱解 $m\colonΩ\subset\mathbb R^2\to\mathbb R^2$ 的映射序列，其熵产生量 $\mathrm{div}\,Φ(m)$ 为 $Ω$ 中的 Radon 测度，受能量控制。此处熵为所有 $C^2$ 向量场 $Φ\colon\mathbb S^1\to\mathbb R^2$，满足 $\mathrm{div}\,Φ(m_*)=0$ 对任何光滑解 $m_*$. 推测熵产生量集中于 $m$ 的一维跳跃集，若 $m$ 具有 bounded variation，则遵循链式法则。特别地，若熵产生量与 $L^p$ 函数一致，则在 $p$ 不太小且自然边界条件下成立。
英文摘要：
The Aviles-Giga energy provides sequences of maps converging to weak solutions $m\colonΩ\subset\mathbb R^2\to\mathbb R^2$ of the eikonal equation \begin{align*} \mathrm{div}\, m=0\text{ in }\mathcal D'(Ω),\quad |m|=1\text{ a.e. in }Ω\,, \end{align*} whose entropy productions $\mathrm{div}\,Φ(m)$ are Radon measures in $Ω$, controlled by the energy. Here, the entropies are all $C^2$ vector fields $Φ\colon\mathbb S^1\to\mathbb R^2$ such that $\mathrm{div}\,Φ(m_*)=0$ for any smooth solution $m_*$. It is conjectured that the entropy production measures are concentrated on the one-dimensional jump set of $m$, as follows from the chain rule if $m$ has bounded variation. In particular, the entropy production measures should vanish if they coincide with $L^p$ functions: this is what we establish in this note if $p$ is not too small and under natural boundary conditions.
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