We shall study the problem of minimizing a functional involving curl of vector fields in a three-dimensional, bounded multiconnected domain with the prescribed tangent component of a given vector field on the boundary. It will be seen that the minimizers are weak solutions of the $p$-curl type system. We shall prove the existence and the estimate of minimizers of a more general functional that contains the $L^p$ norm of the curl of vector fields. We shall also give the continuity with respect to the given data.