On stability of isometric transformations (Q1920802)

A continuous \(W^{1,1}_{\text{loc}} (U)\)-mapping \(f : U \to \mathbb{R}^n\), \(U \subset \mathbb{R}^n\) a domain, is of \(L\)-bounded length distortion (BLD) if for a.e. \(x \in U\), \(J(x,f) \geq 0\) and \(|\xi |/L \leq |f'(x) \xi |\leq L |\xi |\) for all \(\xi \in \mathbb{R}^n\) [\textit{O. Martio} and \textit{J. Väisälä}, Math. Ann. 282, No. 3, 423-443 (1988; Zbl 0632.35021)]. \textit{F. John} [Commun. Pure Appl. Math. 14, 391-413 (1961; Zbl 0102.17404)] showed that if \(f : Q \to \mathbb{R}^n\) is \(L\)-BLD in a cube \(Q \subset \mathbb{R}^n\), then there is an orthogonal matrix \(\theta\) such that for each \(p \geq 1\), \[ \int_Q \bigl |f'(x) - \theta \bigr |^p dx \leq A(L - 1)^p |Q |\tag{*} \] with \(A = A(p,n)\). The author extends this result to John domains, which are domains whose boundary points can be approached from inside by curvelinear cones. An explicit constant \(A\) in (*) is derived for John domains. The proof is different from that of F. John.