Propagation of smallness for solutions of elliptic equations in the plane

Yuzhe Zhu; Yuzhe Zhu

doi:10.3934/mine.2025001

Mathematics in Engineering

2025, Volume 7, Issue 1: 1-12. doi: 10.3934/mine.2025001

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Research article

Propagation of smallness for solutions of elliptic equations in the plane

Yuzhe Zhu ^{†
,
,}

Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
^†Current address: Department of Mathematics, University of Chicago, Chicago, Illinois 60637, USA

Received: 15 December 2024 Revised: 07 January 2025 Accepted: 10 January 2025 Published: 14 January 2025

We explore quantitative propagation of smallness for solutions of two-dimensional elliptic equations from sets of positive $ \delta $-dimensional Hausdorff content for any $ \delta > 0 $. In particular, the gradients of solutions to divergence form equations with Hölder continuous coefficients, as well as those of nondivergence form equations with measurable coefficients, can be quantitatively estimated from the small sets.
- propagation of smallness,
- 2D elliptic equations,
- low regularity coefficients,
- quasiregular mapping,
- spectral inequality
Citation: Yuzhe Zhu. Propagation of smallness for solutions of elliptic equations in the plane[J]. Mathematics in Engineering, 2025, 7(1): 1-12. doi: 10.3934/mine.2025001

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Abstract

We explore quantitative propagation of smallness for solutions of two-dimensional elliptic equations from sets of positive $ \delta $-dimensional Hausdorff content for any $ \delta > 0 $. In particular, the gradients of solutions to divergence form equations with Hölder continuous coefficients, as well as those of nondivergence form equations with measurable coefficients, can be quantitatively estimated from the small sets.

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