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Isoperimetric and Sobolev inequalities for Carnot-Carathéodory spaces and the existence of minimal surfaces. (English) Zbl 0880.35032
The authors develop a general theory of Sobolev and isoperimetric inequalities with the aim of unifying the classical theory due to De Giorgi, Federer, and others and the theory of Carnot-Carathéodory spaces as presented in a recent monograph by M. Gromov [Carnot-Carathéodory spaces seen from within, Prog. Math. 144, 79-323 (1996; Zbl 0864.53025)]. The main object of study are the Sobolev spaces attached to a system of real vector fields \(X=(X_1,\dots,X_m)\) defined on an open set \(\Omega\) in \(\mathbb{R}^n\) and satisfying the Hörmander condition and it is known that these inequalities can play an important role, in particular in the analysis of nonlinear PDEs taking the form \[ \sum^m_{j=1} X^*_j(|Xu|^{p-2} X_ju)= f(x,u,Xu),\quad 1<p<\infty. \] This article contains in addition a very complete list of references on the subject.
Reviewer: B.Helffer (Orsay)

35H10 Hypoelliptic equations
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
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