Universal principles for Kazdan-Warner and Pohozaev-Schoen type identities
Universal principles for Kazdan-Warner and Pohozaev-Schoen type identities
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DOI：
10.1142/s0219199713500028
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发表时间：
2010-10
期刊：
影响因子：
--
通讯作者：
A. Gover;A. Gover;B. Ørsted
中科院分区：
文献类型：
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作者：
A. Gover;A. Gover;B. Ørsted
经典的Pohozaev恒等式约束了一类半线性偏微分方程边值问题的势解。Kazdan-Warner恒等式是球面上共形规定标量曲率的Nirenberg问题的一个类似的必要条件。对于维度$n\geq 3$，两个恒等式都被捕获并扩展为一个恒等式，这是由Schoen在1988年提出的。在这三种情况下，恒等式都要求并涉及到一个无穷小的共形对称。对于具有这样一个共形向量场的结构，我们对这个图进行了一个非常广泛的，基本上是完整的推广。任何共形变分自然标量不变量都被证明满足Kazdan-Warner型恒等式，对于作为局部守恒2张量轨迹的标量也有类似的结果。后一种类型的标量在有边界的流形上也满足Pohozaev-Schoen型恒等式，并有进一步的推广。通过研究全共形变分理论，特别是有关泛函的规范不变性，对这些现象进行了解释和统一。我们对Pohozaev-Schoen恒等式的推广被证明是对物理学和广义相对论中标准守恒定律的补充。
The classical Pohozaev identity constrains potential solutions of certain semilinear PDE boundary value problems. The Kazdan-Warner identity is a similar necessary condition important for the Nirenberg problem of conformally prescribing scalar curvature on the sphere. For dimensions $n\geq 3$ both identities are captured and extended by a single identity, due to Schoen in 1988. In each of the three cases the identity requires and involves an infinitesimal conformal symmetry. For structures with such a conformal vector field, we develop a very wide, and essentially complete, extension of this picture. Any conformally variational natural scalar invariant is shown to satisfy a Kazdan-Warner type identity, and a similar result holds for scalars that are the trace of a locally conserved 2-tensor. Scalars of the latter type are also seen to satisfy a Pohozaev-Schoen type identity on manifolds with boundary, and there are further extensions. These phenomena are explained and unified through the study of total and conformal variational theory, and in particular the gauge invariances of the functionals concerned. Our generalisation of the Pohozaev-Schoen identity is shown to be a complement to a standard conservation law from physics and general relativity.