The set of regular values (in the sense of Clarke) of a Lipschitz map. A sufficient condition for rectifiability of class \(C^3\) (Q2833620)

A Borel subset \(S\) of \(\mathbb R^N\) is said to be \((\mathcal{H}^n,n)\) rectifiable of class \(C^3\) if there exists countably many \(n\)-dimensional submanifolds \(M_j\) of \(\mathbb R^N\) of class \(C^3\) such that \(\mathcal{H}^n\left(S\setminus\bigcup\limits_jM_j\right)=0\). For \(\gamma\in I(n,N)=\{\gamma=(\gamma_1,\dots,\gamma_n)\in\mathbb N^n;\;1\leq\gamma_i<\gamma_{i+1}\leq N\}\) and \(s\in\mathbb R^n\), the Clarke subdifferential \(\partial\varphi^\gamma(s)\) of the map \(\varphi^\gamma=(\varphi^{\gamma_1},\dots,\varphi^{\gamma_n}):\mathbb R^n\to\mathbb R^n\) is defined as NEWLINE\[NEWLINE\partial\varphi^\gamma(s)=\text{co}\left\{\lim\limits_{i\to\infty}D\varphi^\gamma(s_i);\;D\varphi^\gamma(s_i)\;\text{exists},\;s_i\to s\right\}.NEWLINE\]NEWLINE The set \(\partial\varphi^\gamma(s)\) is said to be nonsingular if every matrix in \(\partial\varphi^\gamma(s)\) is of rank \(n\), and \(D\varphi^\gamma(s)\in\partial\varphi^\gamma(s)\) whenever \(\varphi^\gamma\) is differentiable at \(s\).NEWLINENEWLINEIn this paper, the author considers the rectifiability of class \(C^3\) of the set of regular values, in the sense of Clarke, of a Lipschitz map \(\varphi : \mathbb {R}^n\rightarrow \mathbb {R}^N\), \(n<N\). It is proven that if \(c_{1,i},c_{2,i}:\mathbb R^n\to\mathbb R\setminus\{0\}\) is a family of locally bounded functions, \(G_{1,i},G_{2,i}:\mathbb R^n\to\mathbb R^N\), \(H_{ij}:\mathbb R^n\to\mathbb R^N\) are Lipschitz maps, and the set \(A=\{t\in\mathbb R^n\}\) satisfies (i)\, the map \(\varphi\) and all maps \(G_{1,i}\) are differentiable at \(t\), (ii)\, \(D_i\varphi(t)=c_{1,i}(t)G_{1,i}(t)=c_{2,i}(t)G_{2,i}(t)\), (iii)\, \(D_jG_{1,i}(t)=c_{2,j}(t)H_{ij}(t)\), and (iv)\, for almost every \(a\in A\) there exists a nontrivial ball \(B\) centered at \(a\) and such that \(\mathcal{L}^n(B\setminus A)=0\), then \(\varphi(A\cap\mathcal{R})\) is \((\mathcal{H}^n,n)\) rectifiable of class \(C^3\), where \({\mathcal R}=\{s\in\mathbb R^n;\;\partial\varphi^\gamma(s)\;\text{is nonsingular for some}\, \gamma\}\).