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Sharp Gradient Stability for the Sobolev Trace Inequality
Abstract
Let n≥3 and 1<p<n. We prove a quantitative stability estimate for the critical Sobolev trace inequality on the upper half-space. More precisely, the Sobolev trace deficit controls the max{2,p}-th power of the gradient distance to the manifold of trace bubbles. A central part of the proof is the spectral nondegeneracy of the trace bubbles: the first two eigenspaces of the linearized weighted Steklov problem are exactly the amplitude, dilation, and tangential translation modes.
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BibTeX
@misc{ma2026sharp,
title = {{Sharp Gradient Stability for the Sobolev Trace Inequality}},
author = {Xi-Nan Ma and Yitian Zhang and Yang Zhou},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2607.17588},
eprint = {2607.17588},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.17588},
}
RIS
TY - UNPB
TI - Sharp Gradient Stability for the Sobolev Trace Inequality
AU - Xi-Nan Ma
AU - Yitian Zhang
AU - Yang Zhou
PY - 2026
DA - 2026/08/28
PB - arXiv
DO - 10.48550/arxiv.2607.17588
UR - https://arxiv.org/abs/2607.17588
AB - Let n≥3 and 1<p<n. We prove a quantitative stability estimate for the critical Sobolev trace inequality on the upper half-space. More precisely, the Sobolev trace deficit controls the max{2,p}-th power of the gradient distance to the manifold of trace bubbles. A central part of the proof is the spectral nondegeneracy of the trace bubbles: the first two eigenspaces of the linearized weighted Steklov problem are exactly the amplitude, dilation, and tangential translation modes.
ER -