Power type

A nonlinear function $F:\mathbb {R} ^{n}\to \mathbb {R} ^{m}$ is of power type with exponent $p>1$ if one has the bounds

$|F(u)|\leq C|u|^{p}$

(so in particular $F(0)=\nabla F(0)=0$) and

$|F(u)-F(v)|\leq C|u-v|(|u|^{p-1}+|v|^{p-1})$

for all $u,v\in \mathbb {R} ^{n}$ and some constant $C>0$. Note that the first bound is a special case of the second once one assumes that $F(0)=0$. If F is continuously differentiable, then the second bound is equivalent to the bound

$|\nabla F(u)|\leq C|u|^{p-1}$

(possibly for a slightly different value of C.