Abstract
This paper discusses the relaxation of a multiwell energy of the special form W = mini{|▽u − ai|2}. We explain how the relaxation QW can be expressed in terms of certain “tensors of geometric parameters.” The exact set ℱθ of attainable geometric parameters is not known, but we show that it must lie inside an explicitly given convex set ℱ U θ . This leads to a new Geometric Parameters Lower Bound for QW. For the special case of threewells in two space dimensions we give a complete characterization of the extreme points of ℱ U θ . The final section addresses the “three gradient problem,” which asks whether three pairwise incompatible gradients can nevertheless be mutually compatible. We do not solve this problem, but we show that it is linked to the attainability of the type 3 extreme points of ℱ U θ .
The work of NBF was supported in part by the Institute for Mathematics and its Applications with funds provided by the National Science Foundation and the U.S. Army Research Office. An earlier version of this work appeared as part of NBF’s Ph.D. Thesis, Optimal Translations and Relaxations of Some Multiwell Energies, New York University, August 1990.
The work of RVK was supported in part by NSF grant DMS-8701895, ONR grant N00014-88-K-0279, AFOSR grant 90-0090, ARO contract DAAL03-89-K-0039, and DARPA contract F49620-87-C-0065.
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Firoozye, N.B., Kohn, R.V. (1993). Geometric Parameters and the Relaxation of Multiwell Energies. In: Kinderlehrer, D., James, R., Luskin, M., Ericksen, J.L. (eds) Microstructure and Phase Transition. The IMA Volumes in Mathematics and its Applications, vol 54. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8360-4_6
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DOI: https://doi.org/10.1007/978-1-4613-8360-4_6
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