Franchi, Bruno; Serapioni, Raul; Serra Cassano, Francesco Approximation and imbedding theorem for weighted Sobolev spaces associated with Lipschitz continuous vector fields. (English) Zbl 0952.49010 Boll. Unione Mat. Ital., VII. Ser., B 11, No. 1, 83-117 (1997). The authors consider a family of Lipschitz vector fields \(X_1,\dots,X_m\) in an open set \(\Omega\) of \({\mathbb R}^n\), and prove a density result of Meyers-Serrin type in weighted Sobolev spaces associated to \(X_1,\dots,X_m\), when the weight function is in the \(A_p\) class of Muckenhoupt.The results can be applied to the study of the Lavrentieff phenomenon.The case in which it is possible to associate to the vector fields \(X_1,\dots,X_m\) a natural metric \(\rho\) by means of subunit curves is also considered, and a density result is again proved if the weight function belongs to an \(A_p\) class with respect to the metric \(\rho\).The results are applied to prove regularity results for solutions of degenerate elliptic equations, and a Rellich’s type compact imbedding theorem for weighted spaces associated with a family of vector fields. Reviewer: Riccardo De Arcangelis (Napoli) Cited in 56 Documents MSC: 49J45 Methods involving semicontinuity and convergence; relaxation 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 35J70 Degenerate elliptic equations Keywords:weighted Sobolev spaces; vector fields; approximation; weight function; Lavrentieff phenomenon; regularity; degenerate elliptic equations PDFBibTeX XMLCite \textit{B. Franchi} et al., Boll. Unione Mat. Ital., VII. Ser., B 11, No. 1, 83--117 (1997; Zbl 0952.49010)