Nonlocal regularization of L. C. Young's 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.The research of D. Brandon has been partially supported by the Office of Naval Research under Grant Number N00014-88-K-0417 and by DARPA Grant F4920-87-C-0116, and that of R. C. Rogers has been partially supported by the Office of Naval Research under Grant Number N00014-88-K-0417 and by the National Science Foundation under Grant Number DMS-8801412.