Strong monotonicity conjecture for the glacier surface motion map

Let $\textnormal{r}>2$ be such that the Lipschitz continuity conjecture holds, set $\mathcal{X}=W^{1,\textnormal{r}}(\Omega)$, fix $b\in\mathcal{X}$, and define

$$\mathcal{K}=\{r\in\mathcal{X}:r|_{\partial\Omega}=b|_{\partial\Omega}\text{ and }r\geq b\}.$$

Let $\Phi:\mathcal{K}\to\mathcal{X}'$ be the resulting well-defined surface motion map. Strong monotonicity conjecture. There exist constants $\alpha>0$ and $\textnormal{q}>1$ such that

$$\left(\Phi(r)-\Phi(s)\right)[r-s]\geq\alpha\|r-s\|_{\mathcal{X}}^{\textnormal{q}}\qquad\text{for all }r,s\in\mathcal{K}.$$

This coercive monotonicity would provide the key positivity property needed for well-posedness of the backward Euler variational inequality for glacier surface evolution. It is conditional on the preceding Lipschitz conjecture and is not resolved in the paper.