Optimal pinwheel partitions for the Yamabe equation

IF 1.6 2区数学 Q2 MATHEMATICS, APPLIED

Nonlinearity Pub Date : 2024-08-27 DOI:10.1088/1361-6544/ad700c

Mónica Clapp, Jorge Faya, Alberto Saldaña

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引用次数: 0

Abstract

We establish the existence of an optimal partition for the Yamabe equation in

made up of mutually linearly isometric sets, each of them invariant under the action of a group of linear isometries. To do this, we establish the existence of a solution to a weakly coupled competitive Yamabe system, whose components are invariant under the action of the group, and each of them is obtained from the previous one by composing it with a linear isometry. We show that, as the coupling parameter goes to

−∞

, the components of the solutions segregate and give rise to an optimal partition that has the properties mentioned above. Finally, taking advantage of the symmetries considered, we establish the existence of infinitely many sign-changing solutions for the Yamabe equation in

that are different from those previously found by Ding, and del Pino, Musso, Pacard and Pistoia.

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山边方程的最佳针轮分区

我们证明了在由相互线性等距集组成的 RN 中存在山边方程的最优分区，每个等距集在线性等距集群的作用下都是不变的。为此，我们确定了一个弱耦合竞争山边系统解的存在性，该系统的各个部分在该群的作用下都是不变的，而且每个部分都是通过与一个线性等距集组成而从前者得到的。我们的研究表明，当耦合参数为-∞时，解的各组成部分会分离，并产生具有上述性质的最优分区。最后，利用所考虑的对称性，我们确定了山部方程在 RN 中存在无限多个符号变化解，这些解与丁肇中、德尔皮诺、穆索、帕卡德和皮斯托亚之前发现的解不同。

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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来源期刊

Nonlinearity 物理-物理：数学物理

CiteScore

3.00

自引率

5.90%

发文量

170

审稿时长

12 months

期刊介绍： Aimed primarily at mathematicians and physicists interested in research on nonlinear phenomena, the journal''s coverage ranges from proofs of important theorems to papers presenting ideas, conjectures and numerical or physical experiments of significant physical and mathematical interest. Subject coverage: The journal publishes papers on nonlinear mathematics, mathematical physics, experimental physics, theoretical physics and other areas in the sciences where nonlinear phenomena are of fundamental importance. A more detailed indication is given by the subject interests of the Editorial Board members, which are listed in every issue of the journal. Due to the broad scope of Nonlinearity, and in order to make all papers published in the journal accessible to its wide readership, authors are required to provide sufficient introductory material in their paper. This material should contain enough detail and background information to place their research into context and to make it understandable to scientists working on nonlinear phenomena. Nonlinearity is a journal of the Institute of Physics and the London Mathematical Society.