  Variational problems under uniform quasiconvex constraints on the gradient
are studied. In particular, existence of solutions to such problems is proved
as well as existence of lagrange multipliers associated to the uniform
constraint. They are shown to satisfy an Euler-Lagrange equation and a
complementarity property. Our technique consists in approximating the original
problem by a one-parameter family of smooth unconstrained optimization
problems. Numerical experiments confirm the ability of our method to accurately
compute solutions and Lagrange multipliers.
