Existence of solutions to a general geometric elliptic variational problem
Existence of solutions to a general geometric elliptic variational problem
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DOI：
10.1007/s00526-018-1348-4
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发表时间：
2017-04
影响因子：
2.1
通讯作者：
Yangqin Fang;Sławomir Kolasiński
中科院分区：
文献类型：
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作者：
Yangqin Fang;Sławomir Kolasiński
考虑一类非齐次各向异性椭圆泛函在与恒等同伦的光滑变形和局部Hausdorff极限下稳定的闭维子集上的最小化问题。证明了最小值存在于类内，并且是费德勒意义上的可校正集。竞争者的类别编码了跨越边界的概念。我们承认不可纠偏和非紧致的竞争者和边界，并且我们对维数和协维数不做任何限制。一个重要的证明工具是一个新的光滑变形定理。证明的骨架和主要思想遵循Almgren （Ann Math(2) 87:321-391, 1968）的论文。最后我们证明了张成某些闭集的类在同调和上同调意义上满足我们的公理。
We consider the problem of minimising an inhomogeneous anisotropic elliptic functional in a class of closedmdimensional subsets ofwhich is stable under taking smooth deformations homotopic to the identity and under local Hausdorff limits. We prove that the minimiser exists inside the class and is anrectifiable set in the sense of Federer. The class of competitors encodes a notion of spanning a boundary. We admit unrectifiable and non-compact competitors and boundaries, and we make no restrictions on the dimensionmand the co-dimensionother than. An important tool for the proof is a novel smooth deformation theorem. The skeleton of the proof and the main ideas follow Almgren’s (Ann Math (2) 87:321–391, 1968) paper. In the end we show that classes of sets spanning some closed setBin homological and cohomological sense satisfy our axioms.