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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
Citations:
Zbl 0627.00012; Zbl 0331.26013
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