# 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.

##### 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