Lipschitz-continuous local isometric immersions: rigid maps and origami

A rigid map u:Ω⊂Rn→Rm is a Lipschitz-continuous map with the property that at every x∈Ω 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 |u(x)−u(y)|≤|x−y| for every x,y∈Ω; 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 u:Ω⊂R2→R3 (plus a condition which exclude self intersections). If u(Ω)⊂R2 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 Σu 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 Σu of a map u, which is uniquely determined once we fix the value u(x0)∈Rn and the gradient Du(x0)∈O(n) at a single point x0∈Ω∖Σ. We use this characterization to solve a class of Dirichlet problems associated to some partial differential systems of implicit type.