zbMATH — the first resource for mathematics

Approximation and imbedding theorem for weighted Sobolev spaces associated with Lipschitz continuous vector fields. (English) Zbl 0952.49010
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.

49J45 Methods involving semicontinuity and convergence; relaxation
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35J70 Degenerate elliptic equations