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Sharp One-bubble Critical-Point Stability and Global Compactness for the Sobolev Trace Inequality

  • Bao Yu
  • Yang Zhou
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Abstract

Let n≥3 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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Cite

BibTeXRIS
BibTeX
@misc{yu2026sharp,
  title = {{Sharp One-bubble Critical-Point Stability and Global Compactness for the Sobolev Trace Inequality}},
  author = {Bao Yu and Yang Zhou},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2607.21429},
  eprint = {2607.21429},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.21429},
}
RIS
TY  - UNPB
TI  - Sharp One-bubble Critical-Point Stability and Global Compactness for the Sobolev Trace Inequality
AU  - Bao Yu
AU  - Yang Zhou
PY  - 2026
DA  - 2026/08/28
PB  - arXiv
DO  - 10.48550/arxiv.2607.21429
UR  - https://arxiv.org/abs/2607.21429
AB  - Let n≥3 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.
ER  -