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Isoperimetric inequalities for degenerate metrics. (Inégalités isopérimétriques pour des métriques dégénérées.) (French) Zbl 0794.51011

Let \(\Omega\) be an open subset from \({\mathbb{R}}^ n\), \(X_ 1,\ldots,X_ p\) continuous vector fields on \(\Omega\), \(K\) a compact set in \(\Omega\), \(p\) and 2 real numbers, which satisfy the condition \((CW)\) in a metric ball: \[ (r/r_ 0)(u(B_ r)/u(B))^{1/2}\leq a(v(B_ r)/v(B))^{1/p}, (CW) \] and \((u,v)\) a pair of weight functions on \(\Omega\) so that \(v\) satisfies the duplication condition: \[ \mu[B(x,2r)]\leq C\mu[B(x,r)]. (D) \] In Theorem 3 of the article the author states that there is a constant \(C_ K>0\) so that:
i) If \(E\) is a measurable compact part in the metric ball \(B(x',r')\) then \[ u(E)^{1/q}\leq C_ K\liminf_{\epsilon\to 0}[\epsilon^{- 1}v(\{x:p(x,\partial E)<\epsilon\})], \]
ii) If, as another condition, \(E\) is a convex domain with boundary \(E\), then \[ u(E)^{1/q}\leq C_ K(\int_{\partial E}(\sum_ j<X_ j,\eta>^ 2)^{1/2}vdH_{n-1}). \]
[For the special notions see the paper iself].

MSC:

51M16 Inequalities and extremum problems in real or complex geometry
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