Assume \(U\subset \mathbb{R}^n\) is bounded, \(\partial U\) is \(C^1\) and \(p\in [1,\infty)\). Let \(u\in W^{1,p}(U)\). Then

\begin{equation*} u\in W^{1,p}_0(U) \iff Tu=0 \text{ on } \partial U, \end{equation*}

where \(T\) denotes the trace operator . [1, 5.5 Theorem 2]

References Link to heading

1. L. Evans, Partial differential equations. Providence (R. I.): American mathematical society, 1998.