Quadratic growth from a bounded Hessian
= Quadratic growth from a bounded Hessian
{title2=$\|D^2F\|_\infty<\infty\Longrightarrow |F(p)|\leq C(1+|p|^2)$}
For $F\in C^2(\mathbb R^n)$ with bounded <Hessian matrix>, Taylor's formula along the line from zero to $p$ gives $|F(p)|\leq|F(0)|+|DF(0)||p|+\frac12\|D^2F\|_\infty|p|^2$. Its gradient obeys $|DF(p)|\leq|DF(0)|+\|D^2F\|_\infty|p|$. Consequently $\int_\Omega F(Du)$ is finite for $u\in H^1(\Omega)$ when $\Omega$ has finite measure, and $DF(Du)\in L^2$.