Quadratic growth from a bounded Hessian ($\Vert D^{2}F{\Vert }_{\infty }<\infty ⟹|F(p)|\le C(1+|p{|}^2)$)

For $F\in C^2({\mathbb{R}}^n)$ with bounded Hessian matrix, Taylor's formula along the line from zero to $p$ gives $|F(p)|\le |F(0)|+|DF(0)||p|+\frac{1}{2}\Vert D^{2}F{\Vert }_{\infty }|p{|}^2$. Its gradient obeys $|DF(p)|\le |DF(0)|+\Vert D^{2}F{\Vert }_{\infty }|p|$. Consequently ${\int }_{\Omega }F(Du)$ is finite for $u\in H^1(\Omega )$ when $\Omega$ has finite measure, and $DF(Du)\in L^2$.