A penalized formulation for problems with gradient constraints

We extend the penalized formulation for the elastoplastic torsion presented in [FF. Chouly, T. Gustafsson, and P. Hild, IMA J. Numer. Anal. 46 (2026), 2377-2401], to a more general setting that allows to recover at the limit a Monge-Kantorovich mass transport problem. We prove convergence results when the penalty and physical parameters become small. We complete these results with some numerical experiments.