平面障碍问题的全局解分类

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics Pub Date : 2025-04-14 DOI:10.1016/j.aim.2025.110276

Anthony Salib , Georg S. Weiss

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引用次数: 0

摘要

Sakai首先使用复杂分析技术将障碍问题的全局解完全分类为二维。虽然复分析方法在二维上给出了非常简洁的证明，但它没有解决高维的情况，甚至没有解决与二维密切相关的问题。在酒井发表他的证明40年后，Eberle， Figalli和Weiss最近给出了n≥3维的完整分类。本文借鉴Eberle， Figalli和Weiss最近的证明精神，给出了Sakai无界重合集的分类结果的证明。特别是，我们的方法避免了对复杂分析技术的需要，并为复杂分析无法解决的二维问题提供了新的视角。

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Classification of global solutions to the obstacle problem in the plane

Global solutions to the obstacle problem were first completely classified in two dimensions by Sakai using complex analysis techniques. Although the complex analysis approach produced a very succinct proof in two dimensions, it left the higher dimensional cases, and even closely related problems in two dimensions, unresolved. A complete classification in dimensions

n \geq 3

was recently given by Eberle, Figalli and Weiss, forty years after Sakai published his proof. In this paper we give a proof of Sakai's classification result for unbounded coincidence sets in the spirit of the recent proof by Eberle, Figalli and Weiss. Our approach, in particular, avoids the need for complex analysis techniques and offers new perspectives on two-dimensional problems that complex analysis cannot address.

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来源期刊

Advances in Mathematics 数学-数学

CiteScore

2.80

自引率

5.90%

发文量

497

审稿时长

7.5 months

期刊介绍： Emphasizing contributions that represent significant advances in all areas of pure mathematics, Advances in Mathematics provides research mathematicians with an effective medium for communicating important recent developments in their areas of specialization to colleagues and to scientists in related disciplines.