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Research Article | Open Access

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

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Abstract

We explore quantitative propagation of smallness for solutions of two-dimensional elliptic equations from sets of positive δ-dimensional Hausdorff content for any δ > 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.

Keywords

propagation of smallness2D elliptic equationslow regularity coefficientsquasiregular mappingspectral inequality

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Mathematics in Engineering
Volume 7 Issue 1,
February 2025
Pages 1-12
DOI: 10.3934/mine.2025001

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Cite this article:
Zhu Y. Propagation of smallness for solutions of elliptic equations in the plane. Mathematics in Engineering, 2025, 7(1): 1-12. https://doi.org/10.3934/mine.2025001

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Received: 15 December 2024
Revised: 07 January 2025
Accepted: 10 January 2025
Published: 15 February 2025
©2025 the Author(s), licensee AIMS Press.

This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)