Whitney property in two dimensional Sobolev spaces

Bucur, Dorin; Giacomini, Alessandro; Trebeschi, Paola

doi:10.1090/S0002-9939-08-09366-0

Whitney property in two dimensional Sobolev spaces
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by Dorin Bucur, Alessandro Giacomini and Paola Trebeschi PDF

Proc. Amer. Math. Soc. 136 (2008), 2535-2545 Request permission

Abstract:

For $p >1$, we prove that all the functions of $W_\textrm {loc}^{2,p}(\mathbb {R}^2)$ satisfy the Whitney property; i.e., if $u \in W_\textrm {loc}^{2,p}(\mathbb {R}^2)$ is such that $\nabla u=0$ (in the sense of capacity) on a connected set $K\subseteq \mathbb {R}^2$, then $u$ is constant on $K$.

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