embedding of two spaces

Question

Can it be said that $(H_0^1(\Omega)\bigcap L^{\infty}(\Omega), ||..||_{1,2})$ can be continuously embedded in $(C(\Omega),||..||_{\infty})$, where $\Omega$ is a bounded subset of $\textbf{R}^n$, $n \geq 1$?.

Answer 1

Do you know the following result?

If $\Omega$ is bounded, $f \in H^1_0(\Omega)$, $\Phi : \mathbb R \to \mathbb R$ is bounded and Lipschitz, and $\Phi(0) = 0$, then $\Phi \circ f \in H^1_0(\Omega)$.

For instance, if $\alpha > 0$ is sufficiently small and $f(x) = |x|^{-\alpha} - 1$, then $f \in H^1_0(B(0,1))$ (at least when $n \ge 3$ -- you need a somewhat different example when $n=2$). In light of the result above, $\sin(f) \in H^1_0(B(0,1))$, but $\sin(f)$ is not continuous at the origin.