有界区域上Yamabe泛函在泡泡构型附近的Łojasiewicz–Simon不等式
Łojasiewicz--Simon inequalities near bubbling configurations for the Yamabe functional on bounded domains
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中文总结 AI 辅助
该研究针对有界区域上Yamabe泛函的泡泡构型，建立了Łojasiewicz–Simon型梯度不等式，分离参数与余项并获定量估计，为相关抛物流动力学研究提供关键基础。
中文摘要 AI 辅助
我们针对有界光滑区域上的Yamabe泛函，在包含有限多个集中泡泡的构型（带或不带正则分量）附近，建立了Łojasiewicz–Simon型梯度不等式。通过加权分解，将有限维谱参数和泡泡参数与无限维强制余项分离。我们在正则加泡泡及纯泡泡两种 regime 中，获得了参数、余项及对应能隙的定量估计。这些不等式量化了与这类曲面的偏差，对研究相关抛物流的动力学至关重要。
英文摘要
We establish Łojasiewicz--Simon type gradient inequalities for the Yamabe functional on bounded smooth domains near configurations consisting of finitely many concentrating bubbles, with or without a regular component. A weighted decomposition separates the finite-dimensional spectral and bubble parameters from an infinite-dimensional coercive remainder. We obtain quantitative estimates for the parameters, the remainder, and the corresponding energy gaps, in both the regular-plus-bubbling and the pure-bubbling regimes. These inequalities quantify the deviation from such surfaces and are essential for studying the dynamics of related parabolic flows.
