AI 中文总结
研究 Sobolev 迹不等式，先证 Liu 和 Zhang 定理的局部迹类似物，再建立临界迹泛函的 Struwe 型紧致性定理，结合两者得出尖锐的定量单泡临界点稳定性定理。
AI 中文摘要
设 \(n\geq3\) 且 \(1<p<n\)。首先证明了 Liu 和 Zhang 尖锐单泡临界点稳定性定理的局部迹类似物：在正迹泡附近，欧拉 - 拉格朗日余项以尖锐幂次 \(\max\{1,p - 1\}\) 控制到归一化迹泡流形的梯度距离。然后建立了临界迹泛函的 Struwe 型紧致性定理，给出了 Mercuri - Willem 分解的迹对应。结合 Struwe 型紧致性与局部稳定性估计得到尖锐的定量单泡临界点稳定性定理。
英文摘要
Let $n\ge3$ and $1<p<n$. We first prove the local trace analogue of the sharp one-bubble critical-point stability theorem of Liu and Zhang~\cite{LiuZhang2025}: near a positive trace-bubble, the Euler--Lagrange residual controls the gradient distance to the normalized trace-bubble manifold with the sharp power $\max\{1,p-1\}$. Then, we establish a Struwe-type compactness theorem for the critical trace functional, which gives the trace counterpart of the Mercuri--Willem decomposition~\cite{MercuriWillem2010}. Combining Struwe-type compactness with the local stability estimate yields a sharp quantitative one-bubble critical-point stability theorem.
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