Uniqueness of weighted Sobolev spaces with weakly differentiable weights

Details Uniqueness of weighted Sobolev spaces with weakly differentiable weights Uniqueness of weighted Sobolev spaces with weakly differentiable weights

2011-10-13

arxiv.org/abs/1110.2888v1

Mathematics

Functional Analysis

20 pp

Scientific paper

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 \[H^{1,p}_0(\mathbbm{R}^d,w\,dx)=H^{1,p}(\mathbbm{R}^d,w\,dx)=W^{1,p}(\mathbbm{R}^d,w\,dx).\] If $w$ admits a (weak) logarithmic derivative $\nabla w/w$ which is in $L^q_{\textup{loc}}(w\,dx;\mathbbm{R}^d)$, 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 Poincar\'e 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.

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