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On the Clarke’s generalized Jacobian. (English) Zbl 0659.26008
Abstract analysis, Proc. 14th Winter Sch., Srnî/Czech. 1986, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 14, 305-307 (1987).
[For the entire collection see Zbl 0627.00012.]
Let f: \(R^ n\to R^ k\) be a locally Lipschitz mapping. According to the Rademacher’s theorem there exists a set \(E_ 0\) in \(R^ n\) of n- dimensional Lebesgue measure zero such that the Gateaux (i.e. Fréchet) derivative Df(y) exists whenever \(y\in R^ n\setminus E_ 0.\) Using this fact Clarke has introduced the generalized Jacobian \(\partial f(x)\) as the closed convex hull of all possible limits \(\lim_{y_ i\to y}Df(y_ i),\) where \(y_ i\in R^ n\setminus E_ 0.\) Similarly, if \(E_ 0\) is replaced by a null set \(E\subset R^ n\) containing \(E_ 0\), one can define \(\partial_ Ef(x)\). Thus \(\partial_{E_ 0}f(x)=\partial f(x).\) For \(k=1\) Clarke has shown that \(\partial_ Ef(x)=\partial f(x)\) for any null set E containing \(E_ 0\). In Pac. J. Math. 64, 97-102 (1976; Zbl 0331.26013) F. H. Clarke he asked if the same is true when \(k>1\). The note under review answers this question affirmatively. The proof is based on the Gauss-Green theorem.
Reviewer: M.Fabian

MSC:
26B10 Implicit function theorems, Jacobians, transformations with several variables
49J50 Fréchet and Gateaux differentiability in optimization
58C20 Differentiation theory (Gateaux, Fréchet, etc.) on manifolds
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