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Let $g$ be a smooth Riemannian metric on $\mathbb{R}^d$. Let $D=D^k \subseteq \mathbb{R}^d$ be the $k$-dimensional closed unit disk. ($k<d$).

Suppose we are given a $W^{2,2}$ isometric immersion $F:(D,g|_D) \to (\mathbb{R}^d$,e) where $e$ is the Euclidean metric; $F \in W^{2,2}(D,\mathbb{R}^d)$ and $dF_p \in O(T_pD,\mathbb{R}^d)$ a.e.

Let $ND \subseteq T\mathbb{R}^d|_D$ be the normal bundle of $D$ w.r.t to the metric $g$.

Question: Does there exist $q\in W^{1,2}(D;ND^*\otimes\mathbb{R}^d)$ such that $$ dF \oplus q \in \text{SO}(g,e) \, \,? $$ i.e we require $dF_p \oplus q_p:(T_p\mathbb{R}^d,g_p) \to (\mathbb{R}^d,e)$ to be an orientation-preserving isometry almost eveywhere on $D$.

In codimension $d-k=1$ the answer is positive, since $q$ is determined uniquely by $F$ (in a way that preserve the regularity).

In higher codimension some choices have to be made.

Motivation:

This mini-problem arises in the context of determining the energy scale of thin elastic sheets ("thin manifolds inside thick manifolds"), via Gamma-convergence. (Elaborating more seems a digression).

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