On the Uniqueness of Minimizers for a Class of Variational Problems with Polyconvex Integrand
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AbstractWe prove existence and uniqueness of minimizers for a family of energy functionals that arises in Elasticity and involves polyconvex integrands over a certain subset of displacement maps. This work extends previous results by Awi and Gangbo to a larger class of integrands. We are interested in Lagrangians of the form L(A, u) = f(A) + H(det A) − F⋅ u. Here the strict convexity condition on f and H have been relaxed to a convexity condition. Meanwhile, we have allowed the map F to be non-degenerate. First, we study these variational problems over displacements for which the determinant is positive. Second, we consider a limit case in which the functionals are degenerate. In that case, the set of admissible displacements reduces to that of incompressible displacements which are measure preserving maps. Finally, we establish that the minimizer over the set of incompressible maps may be obtained as a limit of minimizers corresponding to a sequence of minimization problems over general displacements provided we have enough regularity on the dual problems. We point out that these results do not rely on the direct methods of the calculus of variations.
CitationAwi, R., & Sedjro, M. (2019). On the Uniqueness of Minimizers for a Class of Variational Problems with Polyconvex Integrand. Acta Applicandae Mathematicae. doi:10.1007/s10440-019-00282-0
SponsorsMost of the work presented in this paper was carried out while R.A. was a postdoctoral fellow at the Institute for Mathematics and its Applications during the IMA’s annual program on Control Theory and its Applications. M.S. gratefully acknowledges the support of the King Abdullah University of Science and Technology.
PublisherSpringer Science and Business Media LLC
JournalActa Applicandae Mathematicae