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Variational problems under uniform quasiconvex constraints on the gradientare studied. In particular, existence of solutions to such problems is provedas well as existence of lagrange multipliers associated to the uniformconstraint. They are shown to satisfy an Euler-Lagrange equation and acomplementarity property. Our technique consists in approximating the originalproblem by a one-parameter family of smooth unconstrained optimizationproblems. Numerical experiments confirm the ability of our method to accuratelycompute solutions and Lagrange multipliers.

Existence and Approximability Results for variational problems under uniform constraints on the gradient by power penalty