Determine sharper constants in the strong-monotonicity estimate

Determine the constants in the strong-monotonicity estimate for the nonlinear closure associated with the monotone discontinuous Petrov–Galerkin formulation of nonlinear diffusion, improving or completing the constant comparison with the corresponding result of Cantin and Heuer.

Background

Proposition 3.2 verifies the abstract assumptions required for global reliability and efficiency of the monotone discontinuous Petrov–Galerkin (MDPG) loss function. In item (iii), the proof derives a strong-monotonicity inequality for the nonlinear closure and explicitly tracks constants through several auxiliary estimates and parameter choices.

The authors state that their argument is comparable to an estimate in the earlier work of Cantin and Heuer but that they cannot obtain the constants appearing there. Thus, an unresolved technical problem is to recover those constants or establish sharper, fully explicit constants for the same nonlinear MDPG setting.