A question on density of Lipschitz functions in weighted Sobolev spaces

Question

Recall that for a domain $\Omega\subset \mathbb{R}^n$, the weighted Sobolev space $W^{1,n}(\Omega,\mu)$ is defined as $f\in L^n(\Omega,\mu)$ and the weak derivative $Df\in L^n(\Omega,\mu)$.

Let now $d\mu(x)=w(x)dx$, where $w:\mathbb{R}^n\to [0,1]$ is a weight. We also assume that $w^{-\frac{1}{n-1}}\in L^1(\Omega,dx)$, where $dx$ represent the Lebesgue measure. I have a continuous function $f\in W^{1,1}(\Omega,dx)$ and I know that $Df\in L^n(\Omega,\mu)$ (so in particular $f\in W^{1,n}(\Omega,\mu)$ as well). I wonder is it ture that there exists a sequence of Lipschitz continuos functions $f_i$ such that $f_i\to f$ locally uniformly and that $|Df_i|\to |Df|$ in $L^n(\Omega)$?

I know the answer is positive if $\mu$ is an $A_p$-weight. I wonder any known weaker condition ensures the above conclusion. Thanks for all possible references on this result.

Answer 1

The problem is not easy to tackle: a conjecture by De Giorgi was that $e^{w}+ e^{w^{-1}} \in L^1(\mathbb{R}^n)$ is a sufficient condition for the coincidence for general $p$. Recently, in 2013, V.V.Zhikov proved this conjecture for $p=2$, with the more general condition: $\mu=w_1w_2 \mathcal{L}^n$, with $w_2$ an $A_2$-weight and $w_1$ satisfies the condition $$ \liminf_{m \to \infty} \frac 1{m^2}\left( \int_{\mathbb{R}^n} w_1^m dx \right)^{1/m} \left( \int_{\mathbb{R}^n} w_1^{-m} dx \right)^{ -1/m} < \infty.$$

The proof is not very long and in fact quite robust. In fact, more recently, there have been two generalization of this result: one by Surnachev, dealing with the general $p>1$ case and also with Sobolev spaces with variable exponent, while the other by Ambrosio, Pinamonti and Speight, which generalize this result even to the metric setting (with general $p>1$).

References:

Zhikov "Density of smooth functions in weighted Sobolev spaces" http://link.springer.com/article/10.1134/S1064562413060173

Surnachev "Density of smooth functions in weighted Sobolev spaces with variable exponent" http://link.springer.com/article/10.1134/S1064562414020045

Ambrosio, Pinamonti, Speight "Weighted Sobolev Spaces on Metric Measure Spaces" http://arxiv.org/abs/1406.3000

Answer 2

I believe it should be enough to assume that your measure $\mu$ supports a weak $(1,p)$-Poincaré inequality and is doubling. Then any $W^{1,p}$ function can be approximated by Lipschitz functions in the corresponding Sobolev norm.

I do not know a reference for this in the context of the Euclidean space, but for the theory in metric spaces, see Theorem 4.1 in

http://www.uam.es/departamentos/ciencias/matematicas/ibero/16.2/Shanmu.pdf.

In a metric space the Sobolev space has to be replaced by a so called Newtonian Space. But this is a real generalization, so if one works in $\mathbb R^n$, they should coincide. A further good reference is the book http://www.ems-ph.org/books/book.php?proj_nr=141.