NONLOCAL REGULARIZATION OF YOUNG.L.C. TACKING PROBLEM

L. C. Young's tacking problem is a prototype of a nonconvex variational problem for which minimizing sequences for the energy do not attain a minimum. The "minimizer" of the energy is usually described as a Young-measure or generalized curve. In many studies, the tacking problem is regularized by adding a higher-order "viscosity term" to the energy. This regularized energy has classical minimizers. In this paper we regularize instead with a spatially nonlocal term. This weakly regularized problem still has measure-valued minimizers, but as the nonlocal term becomes stronger, the measure-valued solutions organize, coalesce, and eventually turn into classical solutions. The information on the measure-valued solutions is obtained by studying equivalent variational problems involving moments of the measures.