Abstract
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.
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Communicated by David Kinderlehrer
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.
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Brandon, D., Rogers, R.C. Nonlocal regularization of L. C. Young's tacking problem. Appl Math Optim 25, 287–301 (1992). https://doi.org/10.1007/BF01182325
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DOI: https://doi.org/10.1007/BF01182325