Published Paper

Inserted: 22 apr 2024
Last Updated: 22 apr 2024

Year: 2022

Abstract:

In this paper we consider an abstract Wiener space $(X,\gamma,H)$ and an open subset $O\subseteq X$ which satisfies suitable assumptions. For every $p\in(1,+\infty)$ we define the Sobolev space $W_{0}^{1,p}(O,\gamma)$ as the closure of Lipschitz continuous functions which support with positive distance from $\partial O$ with respect to the natural Sobolev norm, and we show that under the assumptions on $O$ the space $W_{0}^{1,p}(O,\gamma)$ can be characterized as the space of functions in $W^{1,p}(O,\gamma)$ which have null trace at the boundary $\partial O$, or, equivalently, as the space of functions defined on $O$ whose trivial extension belongs to $W^{1,p}(X,\gamma)$.

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