*Published Paper*

**Inserted:** 6 feb 2008

**Last Updated:** 14 dec 2012

**Journal:** J. Math. Pures Appl.

**Volume:** 90

**Number:** 1

**Pages:** 66-81

**Year:** 2008

**Abstract:**

A *rigid map* $u\colon \Omega \subset \mathbb{R}^{n}\rightarrow
\mathbb{R}^{m}$ is a *Lipschitz-continuous map* with the property that
at every $x\in \Omega $ where $u$ is differentiable then its gradient $Du(x)$
is an *orthogonal* $m\times n$ matrix. If $\Omega$ is
convex, then $u$ is globally a *short map*, in the sense that $\lvert u(x)-u(y)
\rvert\leq
\lvert x-y\rvert$ for every $x,y\in \Omega $; 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 $C^{1}$) 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 $C^{1}$ rigid map $u\colon \Omega \subset
\mathbb{R}^{2}\rightarrow \mathbb{R}^{3}$ (plus a condition which exclude
self intersections). If $u\left( \Omega \right) \subset \mathbb{R}^{2}$ 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 $\Sigma _{u}$ of the gradient
$Du$ of a piecewise $C^{1}$ 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 $\Sigma $ which satisfies the
Kawasaki condition is in fact the *singular set* $\Sigma _{u}$ of a
map $u$, which is uniquely determined once we fix the value $u(x_{0})\in
\mathbb{R}^{n}$ and the gradient $Du(x_{0})\in O(n)$ at a single point $%
x_{0}\in \Omega \setminus \Sigma $. We use this characterization to solve a
class of *Dirichlet problems* associated to some *partial
differential systems* of *implicit type*.

**Keywords:**
rigid maps, origami

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