谐函数节点体积的近似尖锐下界

IF 3.1 1区数学 Q1 MATHEMATICS

Communications on Pure and Applied Mathematics Pub Date : 2024-05-29 DOI:10.1002/cpa.22207

Alexander Logunov, Lakshmi Priya M. E., Andrea Sartori

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引用次数: 0

摘要

本文主要研究谐函数的增长与其零集的 Hausdorff 度量之间的关系。设是一个实值谐函数，且。我们证明了翻倍指数是由定义的增长概念，这给出了、的零集的 Hausdorff 度量的一个近乎尖锐的下限，猜想它是线性的。文章的新内容是稳定增长的概念，以及谐函数倍指数分布下界的多尺度归纳技术。与之前最著名的下界，即纳迪拉什维利猜想相比，它给出了一个重大改进。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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Almost sharp lower bound for the nodal volume of harmonic functions

This paper focuses on a relation between the growth of harmonic functions and the Hausdorff measure of their zero sets. Let be a real‐valued harmonic function in with and . We prove where the doubling index is a notion of growth defined by This gives an almost sharp lower bound for the Hausdorff measure of the zero set of , which is conjectured to be linear in . The new ingredients of the article are the notion of stable growth, and a multiscale induction technique for a lower bound for the distribution of the doubling index of harmonic functions. It gives a significant imuprovement over the previous best‐known bound , which implied Nadirashvili's conjecture.

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来源期刊

Communications on Pure and Applied Mathematics 数学-数学

CiteScore

6.70

自引率

3.30%

发文量

审稿时长

>12 weeks

期刊介绍： Communications on Pure and Applied Mathematics (ISSN 0010-3640) is published monthly, one volume per year, by John Wiley & Sons, Inc. © 2019. The journal primarily publishes papers originating at or solicited by the Courant Institute of Mathematical Sciences. It features recent developments in applied mathematics, mathematical physics, and mathematical analysis. The topics include partial differential equations, computer science, and applied mathematics. CPAM is devoted to mathematical contributions to the sciences; both theoretical and applied papers, of original or expository type, are included.