Quartic gradient monotonicity inequality ($(|\xi {|}^2\xi -|\eta {|}^2\eta )\cdot (\xi -\eta )\ge \frac{1}{4}|\xi -\eta {|}^4$)

For real vectors $\xi ,\eta$, put $m=(\xi +\eta )/2$ and $d=\xi -\eta$. Expansion givesThis 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.