0x68e62b3c

Let \( \Omega \) be bounded with \( C^{1} \)-boundary. By Morrey’s inequality ,

\[ W^{1,\infty}(\Omega) \hookrightarrow C^{0,1}(\overline{\Omega}). \]

In particular, \( u \in W^{1,\infty}(\Omega) \) has a classical trace

\[ u|_{\partial \Omega} \in C(\partial \Omega), \]

and there holds

\[ \|u|_{\partial \Omega}\|_{C(\partial \Omega)} \le \|u\|_{C^{0,1}(\overline{\Omega})} \le C \|u\|_{W^{1,\infty}(\Omega)}, \]

for some constant \( C > 0 \) not depending on \( u \). Thus,

\[ W^{1,\infty}(\Omega) \hookrightarrow C(\partial \Omega). \]