Weighted Sobolev spaces

The case when smooth functions are not dense in a weighted Sobolev space $W$ is considered. New examples of the inequality $H\ne W$ (where $H$ is the closure of the space of smooth functions) are presented. We pose the problem of 'viscosity' or 'attainable' spaces $V$ (that is, spaces that are in a certain sense limits of weighted Sobolev spaces corresponding to 'well-behaved' weights, which means weights bounded above and away from zero) such that $H\subseteq V\subseteq W$. A precise definition of this property of 'attainability' is given in terms of the convergence of the solutions of the corresponding elliptic equations. It is proved that an attainable space always exists, but does not in general coincide with the extreme spaces $H$ and $W$. Examples of strict inclusions $H\subset V\subset W$ are presented.