Regularity of polyharmonic maps in the critical dimension
Regularity of polyharmonic maps in the critical dimension
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DOI：
10.4310/cag.2009.v17.n2.a2
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发表时间：
2009
影响因子：
0.7
通讯作者：
A. Gastel;Christoph Scheven
中科院分区：
文献类型：
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作者：
A. Gastel;Christoph Scheven
近年来，高阶几何变分问题引起了人们的广泛关注。分析师对它们感兴趣的一个共同特征是，它们往往与具有临界增长非线性的高阶偏微分方程组有关。对于这样的偏微分方程，弱解的正则性是一个问题，因为我们处于经典方法无法应用的边缘情况。对于黎曼流形之间的映射u：m→N，到目前为止，经典的变分问题是与能量E(U)：=1 2∫M|Du|2有关的问题，其临界点是调和映射。关于调和映射的正则性问题，尽管有一些未解决的问题，但人们已经很好地理解了。在(区域)的二维中，能量是共形不变的，调和映射是光滑的。对于高维区域，只有部分正则性成立，并且仅对于相对于区域中的变化是稳定的调和映射。因此，在多个维度中，为了产生平滑的极小化，最小化能量似乎不是最好的选择。这就是引入p>；1的p-∫M|Du|p的主要原因。极小子是c1，α，只要p≥n，但对于p=2，由于泛函的非二次增长，这不能改进为给出C∞。此外，EP临界点的正则性在一般情况下仍是一个未解决的问题。为了得到更自然的具有二次增长的变分问题，高阶泛函似乎是一个很好的选择。已经有了一个快速的
Higher order geometric variational problems have attracted quite some attention in recent years. A common feature that makes them interesting for the analyst is the fact that they tend to be associated to systems of higher order partial differential equations with critical growth nonlinearities. For such partial differential equations, regularity of weak solutions is an issue, since we are in a borderline case where classical methods just fail to apply. For mappings u : M → N between Riemannian manifolds, the by now classical variational problem is the one associated to the energy E(u) := 1 2 ∫ M |Du|2, the critical points of that are harmonic maps. Regularity questions for harmonic maps are quite well understood, in spite of some open questions. In two dimensions (of the domain) the energy is conformally invariant, and harmonic maps are smooth. For higher-dimensional domains only partial regularity holds, and only for harmonic maps that are stationary with respect to variations in the domain. Therefore, in more than two dimensions, minimizing the energy does not seem to be the best choice in order to produce smooth minimizers. This is the main reason why the p-energy Ep(u) := 1 p ∫ M |Du|p for p > 1 has been introduced. Minimizers are C1,α as long as p ≥ n, but for p = 2 this cannot be improved to give C∞, due to the non-quadratic growth of the functional. Moreover, the regularity for critical points of Ep is still an open problem in the general case. In order to get more natural variational problems with quadratic growth, higher order functionals seem to be a good choice. There has been a quickly