Represent $\partial \Omega$ as an embedding $F:\mathbb S^2 \to \mathbb R^3$ and consider the mean curvature flow $\{F_t :\mathbb S^2 \to \mathbb R^3: t\in [0,T)\}$, which is a family of embeddings so that
$$\frac{\partial F}{\partial t} = H_t v_t,.$$
where $v_t$ is the inward unit normal of the embedding $F_t$. Since $\mathbb S^2$ is compact, the solution exist for short time $T$. The evolution of $H$ is given by
$$\frac{\partial H}{\partial t} = \Delta H + |A|^2 H.$$
The strong maximum principle implies that $H>0$ when $t>0$.