= Quartic gradient monotonicity inequality
{title2=$( |\xi|^2\xi-|\eta|^2\eta)\cdot(\xi-\eta)\geq\tfrac14|\xi-\eta|^4$}
For real <vectors> $\xi,\eta$, put $m=(\xi+\eta)/2$ and $d=\xi-\eta$. Expansion gives
$$
(|\xi|^2\xi-|\eta|^2\eta)\cdot(\xi-\eta)
=|m|^2|d|^2+2(m\cdot d)^2+\frac14|d|^4
\geq\frac14|d|^4.
$$
This proves strict <operator monotonicity> of the <gradient> of $\xi\mapsto|\xi|^4/4$. Integrating the inequality controls the difference of two <weak gradients> in the fourth power of their <Lp norm>, which yields uniqueness for suitable nonlinear <Dirichlet boundary conditions>.
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