Uniqueness of weighted Sobolev spaces with weakly differentiable weights

Abstract

We prove that weakly differentiable weights $w$ which, together with their reciprocals, satisfy certain local integrability conditions, admit a unique associated first-order $p$-Sobolev space, that is \[H^{1,p}(\mathbb{R}^d,w\,\d x)=V^{1,p}(\mathbb{R}^d,w\,\d x)=W^{1,p}(\mathbb{R}^d,w\,\d x),\] where $d\in \mathbb{N}$ and $p\in [1,\infty )$. If $w$ admits a (weak) logarithmic gradient $\nabla w/w$ which is in $L^q_{\text{loc}}(w\,\d x;\R^d)$, $q=p/(p-1)$, we propose an alternative definition of the weighted $p$-Sobolev space based on an integration by parts formula involving $\nabla w/w$. We prove that weights of the form $\exp (-\beta |\cdot {|}^q-W-V)$ are $p$-admissible, in particular, satisfy a Poincaré inequality, where $\beta \in (0,\infty )$, $W$, $V$ are convex and bounded below such that $|\nabla W|$ satisfies a growth condition (depending on $\beta$ and $q$) and $V$ is bounded. We apply the uniqueness result to weights of this type. The associated nonlinear degenerate evolution equation is also discussed.

arXiv comment23 pp., to appear in J. Funct. Anal. (in press)

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