Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^n$. If I consider a function $g:\mathbb{R}\rightarrow\mathbb{R}$ which has the following properties: $$ |g(x)|\leq M \qquad |g(x)-g(y)|\leq K|x-y| $$ the function $u\in H^1(\Omega)$ and the composite function $g(u)$, can I say that: $$ \|g(u)\|_{H^1(\Omega)}\leq\sqrt{|\Omega|(M^2+K^2)} \quad? $$
Another question: is it possible to use this disequation? $$ \|g(u)\|_{L^2(\partial\Omega)}\leq C_{tr}\|g(u)\|_{H^1(\Omega)} $$