集中现象与消失质量猜想
Concentration phenomena and the Vanishing Mass Conjecture
- University of Warwick（华威大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文针对所有一阶常系数线性微分算子完全解决了Bouchitté提出的消失质量猜想，给出了集中序列的完整结构描述，并获得了多个相关应用结果。
AI中文摘要：
满足线性偏微分方程（PDE）约束的集中（即非等可积）函数序列，出现在PDE、变分法和几何测度论的众多问题中。2003年，Bouchitté提出猜想：此类集中现象总能表示为“简单”集中的叠加，他将该主张称为消失质量猜想。我们针对所有一阶常系数线性微分算子完全解决了该猜想，证明任意此类序列的值分布（Young）测度可接受Choquet型分解，其概率测度的重心位于相关的Tartar波锥中。事实上，我们证明了比原猜想更强的结论，即组成测度本身由满足相同PDE约束的集中序列生成。这为集中序列提供了完整的结构描述，并产生若干应用，包括De Philippis与第二作者关于PDE约束测度奇异极定理的新证明、两类补偿紧致性结果、与形状优化中最优光结构猜想相关的一般下界，以及关于极值集中Young测度支撑基数的意外结果。我们还将结果置于Morrey猜想的框架下，表明其针对纯集中的类似命题不成立。我们的证明引入了若干新技术，最显著的是涉及“障碍函数”的凸性论证，以及通过热流进行的精细平滑过程，该过程替代了经典的傅里叶方法。
英文摘要：
Concentrating (that is, non-equi-integrable) sequences of functions satisfying linear PDE constraints arise in a wide variety of problems in PDE, the calculus of variations, and geometric measure theory. In 2001, Bouchitté conjectured that such concentrations can always be represented as superpositions of ``simple'' concentrations, a claim he termed the Vanishing Mass Conjecture. We fully resolve this conjecture for all first-order constant-coefficient linear differential operators, proving that the value-distribution (Young) measure of any such sequence admits a Choquet-type decomposition into probability measures whose barycenters lie in the associated Tartar wave cone. In fact, we prove a significantly stronger statement than originally conjectured, namely that the constituent measures are themselves generated by concentrating sequences satisfying the same PDE constraint. This provides a complete structural description of concentrating sequences and yields several applications, including a new proof of a theorem of De~Philippis and the second author on the singular polar of PDE-constrained measures, two types of compensated compactness results, and a surprising result on the support cardinality of extremal concentration Young measures. We also place our results in the context of Morrey's Conjecture, showing that its analogue for pure concentrations is false. Our proofs introduce several new techniques, most notably convexity arguments involving ``barrier functions'' and a careful smoothing procedure via the heat flow that replaces classical Fourier methods.
