Sobolev spaces and harmonic maps for metric space targets
Sobolev spaces and harmonic maps for metric space targets
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DOI：
10.4310/cag.1993.v1.n4.a4
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发表时间：
1993
影响因子：
0.7
通讯作者：
Nicholas J. Korevaar;R. Schoen
中科院分区：
文献类型：
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作者：
Nicholas J. Korevaar;R. Schoen
当人们研究黎曼流形之间的映射的变分问题时，必须考虑我们记为Vr '（r ~ 2，X）的空间。这里ft是黎曼流形中的紧致域，X是第二黎曼流形，p G [l，oo），W表示映射的一阶导数为L（0）。当p > n时，这类映射是连续的，相应的空间W（Cl^X）可以被赋予光滑Banach流形的结构。这是因为，对于p > n，任何在距离上接近映射UQ的映射都可以被描述为UQ的逐点小变形。这个W变形的线性空间是一个Banach空间，可以在其上局部地模拟W '（Q^ X）。对于p {p，，X）变得不那么清楚。这个问题最早是由C. B. Morrey [Mo]在n = dimfi = 2和p = 2的情况下。莫里花了很大的努力来给这个空间下定义。最近，人们利用了J. Nash的嵌入定理，并认为X是欧几里得空间M^的光滑子流形。如果我们将W '（fi，X）定义为Banach空间VF^f^R^）的子集，该空间由图像本质上位于X中的那些映射组成，那么事实证明，这给出了用于许多目的的可行定义。这个定义的一个美学上的缺点是，空间VF '（J ~ 7，X）应该只依赖于X的度量，而不依赖于X嵌入到R中。一个更严重的困难出现，如果一个试图考虑映射到空间X是不光滑的黎曼流形。这些
When one studies variational problems for maps between Riemannian manifolds one must consider spaces which we denote Vr'(r2,X). Here ft is a compact domain in a Riemannian manifold, X is a second Riemannian manifold, p G [l,oo), and W indicates that the first derivatives of the map are L(0). For p > n such maps will be continuous, and the corresponding space W(Cl^X) can be given the structure of a smooth Banach manifold. This is because, for p > n, any map which is close in W^ distance to a map ^o can be described as a pointwise small deformation of UQ. This linear space of W deformations is then a Banach space on which one can locally model W'(Q^ X). For p {p,,X) becomes much less clear. This problem was first encountered by C.B. Morrey [Mo] in case n = dimfi = 2 and p = 2. A great deal of effort was spent by Morrey to give a definition of this space. In more recent times people have exploited the embedding theorem of J. Nash, and considered X to be a smooth submanifold of a Euclidean space M^. If we define W'(fi, X) to be the subset of the Banach space VF^f^R^) consisting of those maps with image essentially in X, it turns out that this gives a workable definition for many purposes. An aesthetic drawback of this definition is that the space VF'(J7, X) should depend only on the metric of X and not on the embedding of X into R. A much more serious difficulty arises if one attempts to consider maps to spaces X which are not smooth Riemannian manifolds. These