Invertibility versus Lagrange equation for traction free energy-minimal deformations
Invertibility versus Lagrange equation for traction free energy-minimal deformations
复制标题
牵引自由能最小变形的可逆性与拉格朗日方程
DOI：
10.1007/s00526-014-0719-8
复制
发表时间：
2015
影响因子：
2.1
通讯作者：
Jani Onninen
中科院分区：
文献类型：
--
作者：
T. Iwaniec;Jani Onninen
设$$\，{\mathbb{X}，{\mathbb{Y}}\子集{\mathbb{R}^2\，$$X，Y⊂R2是具有相同拓扑类型的有界Jordan域，且$$，h：{\mathbb{X}}\xright tarrow[]{{}_{\！\！\mathm{on\，\，}\！}}{\mathbb{Y}}\，$$h：X→ontoY是狄利克雷能量积分的一个无牵引力极小映射.证明了限制在任意子域$$\，{\mathcal{O}}\子集{\mathbb{X}\，$$O→X上的$$，h：{\mathcal{O}}\，$$h：O⊂Y是内射的当且仅当它在$$\，{\mathcal{O}\，$$O中调和。这一结果似乎与其他能量积分有关，并且更一般地，可以解释为，物质的穿透(在薄板的超弹性变形下)在拉格朗日方程失效的地方是不可避免的。
Let $$\,{\mathbb {X}} , {\mathbb {Y}} \subset {\mathbb {R}}^2 \,$$X,Y⊂R2 be bounded Jordan domains of the same topological type and $$\,h : {\mathbb {X}} \xrightarrow []{{}_{\!\!\mathrm{onto\,\,}\!\!}}{\mathbb {Y}}\,$$h:X→ontoY a traction free minimal mapping for the Dirichlet energy integral. It is shown that $$\,h : {\mathcal {O}} \rightarrow {\mathbb {Y}}\,$$h:O→Y, restricted to any subdomain $$\,{\mathcal {O}} \subset {\mathbb {X}}\,$$O⊂X, is injective if and only if it is harmonic in $$\,{\mathcal {O}}\,$$O. This result appears pertinent to other energy integrals and, in greater generality, may be interpreted as saying that the interpenetration of matter (under hyperelastic deformations of thin plates) is inevitable precisely in the localities where the Lagrange equation fails.