Consider the Euclidean space ${\mathbb{R}}^n$ with the orthogonal action of a compact Lie group G. We prove that a locally Lipschitz G-invariant mapping f from ${\mathbb{R}}^n$ to $\mathbb{R}$ can be uniformly approximated by G-invariant smooth mappings g in such a way that the gradient of g is a graph approximation of Clarke’s generalized gradient of f. This result enables a proper development of equivariant gradient degree theory for a class of set-valued gradient mappings.

中文翻译：

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广义梯度等变多值映射，逼近度和度

考虑欧几里德空间 ${\mathbb{[R}}^{ñ}$具有紧Lie群G的正交作用。我们证明了一个局部Lipschitz摹-invariant映射˚F从${\mathbb{[R}}^{ñ}$ 至 $\mathbb{[R}$可以通过均匀地近似ģ -invariant平滑映射克在这样的方式的梯度克是克拉克的广义的梯度的曲线图近似˚F。该结果使一类集值梯度映射的等变梯度度理论得以适当发展。