Lipschitz-continuous local isometric immersions: rigid maps and origami

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• 作者：B. Dacorogna ; P. Marcellini ; E. Paolini
• 关键词：Lipschitz isometric immersions ; Origami ; Dirichlet problem
• 刊名：Journal des Mathematiques Pures et Appliquees
• 出版年：2008
• 出版时间：July 2008
• 年：2008
• 卷：90
• 期：1
• 页码：66-81
• 全文大小：702 K

文摘

A rigid map is a Lipschitz-continuous map with the property that at every xΩ where u is differentiable then its gradient Du(x) is an orthogonal m×n matrix. If Ω is convex, then u is globally a short map, in the sense that 93b5c288a9e2e2b593c4"" title=""Click to view the MathML source"" alt=""Click to view the MathML source"">|u(x)−u(y)||x−y| for every x,yΩ; while locally, around any point of continuity of the gradient, u is an isometry. Our motivation to introduce Lipschitz-continuous local isometric immersions (versus maps of class C¹) is based on the possibility of solving Dirichlet problems; i.e., we can impose boundary conditions. We also propose an approach to the analytical theory of origami, the ancient Japanese art of paper folding. An origami is a piecewise C¹ rigid map (plus a condition which exclude self intersections). If we say that u is a flat origami. In this case (and in general when m=n) we are able to describe the singular set Σ_u of the gradient Du of a piecewise C¹ rigid map: it turns out to be the boundary of the union of convex disjoint polyhedra, and some facet and edge conditions (Kawasaki condition) are satisfied. We show that these necessary conditions are also sufficient to recover a given singular set; i.e., we prove that every polyhedral set Σ which satisfies the Kawasaki condition is in fact the singular set Σ_u of a map u, which is uniquely determined once we fix the value 3b29""> and the gradient b2887e854dd3e"" title=""Click to view the MathML source"" alt=""Click to view the MathML source"">Du(x₀)O(n) at a single point x₀ΩΣ. We use this characterization to solve a class of Dirichlet problems associated to some partial differential systems of implicit type.