关于 $f\mapsto \int \det \nabla f$ 弱不连续性的一个简单例子
A simple example of the weak discontinuity of $f\mapsto \int \det \nabla f$
AI总结：
本文通过一个已知的共形微分同胚弱收敛序列，说明即使将函数空间限制为性质尽可能好的函数，行列式积分泛函仍会表现出弱不连续性。
摘要：
验证积分泛函在 Sobolev 空间弱拓扑下的下半连续性是变分法的核心主题。对于具有 $p$-增长的积分泛函，拟凸性是 $W^{1,p}$ 中弱下半连续性的必要条件，但只有在满足某些附加条件时才是充分条件。表明附加条件必要性的标准泛函是 $f\mapsto \int_Ω\det \nabla f$，它不具有弱下半连续性。然而，展示这一失效的常见例子都是非单射的，并且具有大量剪切。这篇短文的目的是指出，一个已知的 $d$ 维单位球的共形微分同胚序列在 $W^{1,d}$ 中弱收敛于一个常数，即使将空间限制为“尽可能好”的函数，该序列也体现了该泛函的弱不连续性。
英文摘要：
Verifying lower-semicontinuity of integral functionals in the weak topology of Sobolev spaces is a central theme in the calculus of variations. For integral functionals with $p$-growth, quasiconvexity is a necessary condition for weak lower-semicontinuity in $W^{1,p}$, but is only sufficient if some additional conditions are met.The standard functional showing the necessity of additional conditions is $f\mapsto \int_Ω\det \nabla f$, which fails to be weakly lower-semicontinuous. However, the common examples showing this failure are non-injective and have a lot of shear. The aim of this short note is to point out that a known sequence of conformal diffeomorphisms of the $d$-dimensional unit ball that converges weakly to a constant in $W^{1,d}$, exemplifies the weak discontinuity of this functional even when restricting a space to functions which are "as nice as possible".
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
