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<p>We study various properties of the gradients of solutions to harmonic functions on Lipschitz surfaces. We improve an exponential bound of Naber and Valtorta (Commun Pure Appl Math 70(10):1835–1897, 2017) on the size of the superlevel sets for the frequency function to a sharp quadratic bound in this setting using complex analytic tools. We also develop a propagation of smallness for gradients of harmonic functions, settling an open question from Logunov and Malinnikova (Quantitative propagation of smallness for solutions of elliptic equations, World Scientific Publishing, Hackensack, 2018) in this setting. Finally, we extend the estimate on superlevel sets of the frequency to more general divergence-form elliptic PDEs with bounded drift terms at the cost of a subpolynomial factor.</p>

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Results on gradients of harmonic functions on Lipschitz surfaces

  • 期刊论文

  • 发表日期：

    2025年7月5日

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  • Benjamin Foster

摘要

We study various properties of the gradients of solutions to harmonic functions on Lipschitz surfaces. We improve an exponential bound of Naber and Valtorta (Commun Pure Appl Math 70(10):1835–1897, 2017) on the size of the superlevel sets for the frequency function to a sharp quadratic bound in this setting using complex analytic tools. We also develop a propagation of smallness for gradients of harmonic functions, settling an open question from Logunov and Malinnikova (Quantitative propagation of smallness for solutions of elliptic equations, World Scientific Publishing, Hackensack, 2018) in this setting. Finally, we extend the estimate on superlevel sets of the frequency to more general divergence-form elliptic PDEs with bounded drift terms at the cost of a subpolynomial factor.