Let B denote the open unit ball in $R^n$, $B_+ = \{x \in B : x_n > 0\}$ and $B_- = \{x \in B : x_n < 0\}$. Also Suppose u ∈ $C^1(\overline B_+)\bigcap C^1(\overline B_-)$.
I was trying to figure out is it true that u ∈ $W^{1,∞}(B)$? Is there any counterexample? If yes, then what kind of modification or addition of the assumptions is needed to make the statement correct?
Any kind of help is greatly appreciated. Thanks.
This is not true in general. Recall that $u\in W^{1,\infty}(B)$ implies that $u$ is Lipschitz continuous. We define $u\equiv 1$ on $B_+$ and $u\equiv -1$ on $B_-$, then $u$ is not Lipschitz continuous. I think if your impose the condition that $u$ is continuous on $\{x\in B:\,x_n=0\}$, then it is true.