Invitation to H-systems in higher dimensions: known results, new facts, and related open problems
Invitation to H-systems in higher dimensions: known results, new facts, and related open problems
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邀请更高维度的 H 系统：已知结果、新事实和相关的开放问题
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发表时间：
2016
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通讯作者：
Pawel Strzelecki
中科院分区：
文献类型：
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作者：
A. Schikorra;Pawel Strzelecki
在本文中，我们讨论了两个著名的开放性问题的非线性，共形不变的椭圆系统的维数为$nge 3$，具有临界非线性：$H$-系统（方程的超曲面的规定平均曲率）和$n$-调和映射到紧黎曼流形的正则性理论。
对于$n=2$这些问题的几个解决方案是已知的，但他们都打破了更高的维度（除非考虑特殊情况，例如超曲面的常平均曲率或流形的对称性）。我们讨论了一些已知的证据和暗示的主要困难。
我们还陈述了一些新的结果（例如对于偶数$n$，而不是$W ^{1，n}$，$W ^{n/2，2}$类的所有解的肯定答案），并列出一些独立感兴趣的开放问题-包括Coifman-Rochberg-韦斯定理的特定端点变体，解决分数和奇异积分与$W^{1，n}类的有界函数相乘的有界性，n}$ -这将导致这两个问题的解决方案。
In this paper, we discuss two well-known open problems in the regularity theory for nonlinear, conformally invariant elliptic systems in dimensions $nge 3$, with a critical nonlinearity: $H$-systems (equations of hypersurfaces of prescribed mean curvature) and $n$-harmonic maps into compact Riemannian manifolds.
For $n=2$ several solutions of these problems are known but they all break down in higher dimensions (unless one considers special cases, e.g. hypersurfaces of constant mean curvature or manifolds with symmetries). We discuss some of the known proofs and hint at the main difficulties.
We also state a few new results (such as positive answers for all solutions of class $W^{n/2,2}$ for even $n$, instead of $W^{1,n}$) and list some open questions of independent interest - including specific endpoint variants of the Coifman-Rochberg-Weiss theorem, addressing the boundedness of commutators of fractional and singular integrals with multiplication by bounded functions of class $W^{1,n}$ - that would lead to solutions of these two problems.