A note on the support of a Sobolev function on a 𝑘-cell

Ziemer, W.

doi:10.1090/S0002-9939-03-07335-0

A note on the support of a Sobolev function on a $k$-cell
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by W. K. Ziemer PDF

Proc. Amer. Math. Soc. 132 (2004), 1987-1995 Request permission

Abstract:

It is shown that a $k$-cell (the homeomorphic image of a closed ball in $\mathbb {R}^{k}$) in $\mathbb {R}^{n}$, $1\leq k<n$, cannot support a function in $W^{1,p}(\mathbb {R}^{n})$ if $p>[\frac {k+1}{2}]$, the greatest integer in $(k+1)/2$.

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