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Geometric sets of elliptic equations with Dini coefficients

  • Wenshuai Jiang
  • Junyuan Wang
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Abstract

We study divergence-form elliptic equations with Dini coefficients, extending the frameworks of \cite{NV} for Lipschitz coefficients and \cite{HJ1} for Hölder coefficients. We first establish an almost monotonicity property for the doubling index under merely continuous coefficients. This leads to weak volume estimates for critical sets and uniform volume estimates for nodal sets. The Dini assumption is essential here since we construct an example with continuous coefficients and bounded doubling index whose nodal set has infinite measure. We also prove polynomial growth estimates for sub-level sets. Using a different approach, we establish quantitative uniqueness of tangent maps and a cone-splitting principle, which yield explicit Minkowski-type estimates for critical sets. Finally, combined with the results of Kenig--Zhao \cite{KZ4}, our estimates give measure bounds for boundary critical sets.

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Cite

BibTeXRIS
BibTeX
@misc{jiang2026geometric,
  title = {{Geometric sets of elliptic equations with Dini coefficients}},
  author = {Wenshuai Jiang and Junyuan Wang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.17224},
  eprint = {2609.17224},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.17224},
}
RIS
TY  - UNPB
TI  - Geometric sets of elliptic equations with Dini coefficients
AU  - Wenshuai Jiang
AU  - Junyuan Wang
PY  - 2026
DA  - 2026/09/15
PB  - arXiv
DO  - 10.48550/arxiv.2609.17224
UR  - https://arxiv.org/abs/2609.17224
AB  - We study divergence-form elliptic equations with Dini coefficients, extending the frameworks of \cite{NV} for Lipschitz coefficients and \cite{HJ1} for Hölder coefficients. We first establish an almost monotonicity property for the doubling index under merely continuous coefficients. This leads to weak volume estimates for critical sets and uniform volume estimates for nodal sets. The Dini assumption is essential here since we construct an example with continuous coefficients and bounded doubling index whose nodal set has infinite measure. We also prove polynomial growth estimates for sub-level sets. Using a different approach, we establish quantitative uniqueness of tangent maps and a cone-splitting principle, which yield explicit Minkowski-type estimates for critical sets. Finally, combined with the results of Kenig--Zhao \cite{KZ4}, our estimates give measure bounds for boundary critical sets.
ER  -