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Uniform estimates for elliptic systems with critical Sobolev coefficients

  • Tian Lan
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Abstract

The W1,p estimate for uniformly elliptic operators of divergence form with VMO coefficients is classical. In particular, this provides W1,p estimates for elliptic operators with coefficients in W1,n. However, this theory does not yield constants in the estimates controlled by the W1,n norm of the coefficients, since the total W1,n norm does not control the rate of decay for the mean oscillation. In this work, we establish uniform estimates for divergence-form elliptic systems whose coefficients lie in the critical Sobolev--Lorentz spaces W1,(n,s) with 1≤ s<∞, have an L∞-bounded spatially symmetric part, and satisfy an integral ellipticity condition. The constant in the estimate is controlled by the corresponding norms of coefficients and the ellipticity constant.

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Cite

BibTeXRIS
BibTeX
@misc{lan2026uniform,
  title = {{Uniform estimates for elliptic systems with critical Sobolev coefficients}},
  author = {Tian Lan},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.22366},
  eprint = {2610.22366},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.22366},
}
RIS
TY  - UNPB
TI  - Uniform estimates for elliptic systems with critical Sobolev coefficients
AU  - Tian Lan
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2610.22366
UR  - https://arxiv.org/abs/2610.22366
AB  - The W^1,p estimate for uniformly elliptic operators of divergence form with VMO coefficients is classical. In particular, this provides W^1,p estimates for elliptic operators with coefficients in W^1,n. However, this theory does not yield constants in the estimates controlled by the W^1,n norm of the coefficients, since the total W^1,n norm does not control the rate of decay for the mean oscillation. In this work, we establish uniform estimates for divergence-form elliptic systems whose coefficients lie in the critical Sobolev--Lorentz spaces W^1,(n,s) with 1≤ s<∞, have an L^∞-bounded spatially symmetric part, and satisfy an integral ellipticity condition. The constant in the estimate is controlled by the corresponding norms of coefficients and the ellipticity constant.
ER  -