Monotone Sobolev Mappings of Planar Domains and Surfaces
Monotone Sobolev Mappings of Planar Domains and Surfaces
复制标题
平面域和曲面的单调 Sobolev 映射
DOI：
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发表时间：
2015
影响因子：
2.5
通讯作者：
Jani Onninen
中科院分区：
文献类型：
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作者：
T. Iwaniec;Jani Onninen
Young(Duke Math J 15，87-94,1948)的一个逼近定理断言紧致定向拓扑2-流形(曲面)之间的连续映射是单调的当且仅当它是同胚的一致极限。Sobolev映射的相似逼近是几何函数理论(GFT)和非线性弹性理论(NE)的核心。在这两个理论中，所讨论的映射是作为能量最小化的同胚序列的弱极限自然产生的。结果，能量最小的映射变成了单调的。相反，我们证明了Sobolev空间W1，p，1<；p<；∞文档类[12pt]{Minimum}usepackage{amsath}usepackage{amsFonts}usepackage{amssymb}usepackage{amsbsy}usepackage{upgreek}集长度{oddsidemarin}{-69pt}中的单调映射，如{Document}$$，{fancyscript{W}}^{1，p}，，1&p；Inty，}$$end{Document}，不是别人，正是w1，pDocumentclass[12pt]{Minimal}usepackage{amsath}usepackage{wa ysym}usepackage{amsfonts}usepackage{amsbsy}usepackage{mathsfs}usepackage{upgreek}setlong{oddsidemargin}{-69pt}例如{document}$${，{fancyscript{W}}^{1，p}，}$$end{Document}-同态映射的弱(也强)限制。事实上，这些都是微分同胚的极限。作为说明，我们证明了p-调和型能量积分的牵引力自由能-最小变形的存在性。
An approximation theorem of Youngs (Duke Math J 15, 87–94, 1948) asserts that a continuous map between compact oriented topological 2-manifolds (surfaces) is monotone if and only if it is a uniform limit of homeomorphisms. Analogous approximation of Sobolev mappings is at the very heart of Geometric Function Theory (GFT) and Nonlinear Elasticity (NE). In both theories the mappings in question arise naturally as weak limits of energy-minimizing sequences of homeomorphisms. As a result of this, the energy-minimal mappings turn out to be monotone. In the present paper we show that, conversely, monotone mappings in the Sobolev space W1,p,1<p<∞documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${,{fancyscript{W}}^{1,p},, ,1 < p < infty,}$$end{document}, are none other than W1,pdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${,{fancyscript{W}}^{1,p},}$$end{document}-weak (also strong) limits of homeomorphisms. In fact, these are limits of diffeomorphisms. By way of illustration, we establish the existence of traction free energy-minimal deformations for p -harmonic type energy integrals.