双连通区域间能量极小同态的存在性
Existence of energy-minimal diffeomorphisms between doubly connected domains
摘要：
本文建立了两个平面区域之间的同胚最小化Dirichlet能量的存在性。具体地说，在有界双连通域之间的所有同胚f：R -> R* 使得Mod（R）< Mod（R*）中，存在唯一的直到R的共形自同态的能量最小的自同态。没有边界条件施加在f上。虽然任何能量极小的非同态都是调和的，但我们的结果强调了调和非同态的存在性与能量极小非同态的存在性之间的主要区别。全局可逆能量极小映射的存在性是非线性弹性力学数学模型的主要研究内容，也是计算机图形学的一个研究热点。
英文摘要：
The paper establishes the existence of homeomorphisms between two planar domains that minimize the Dirichlet energy. Specifically, among all homeomorphisms f : R -> R* between bounded doubly connected domains such that Mod (R) < Mod (R*) there exists, unique up to conformal authomorphisms of R, an energy-minimal diffeomorphism. No boundary conditions are imposed on f. Although any energy-minimal diffeomorphism is harmonic, our results underline the major difference between the existence of harmonic diffeomorphisms and the existence of the energy-minimal diffeomorphisms. The existence of globally invertible energy-minimal mappings is of primary pursuit in the mathematical models of nonlinear elasticity and is also of interest in computer graphics.
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