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    On the Uniqueness of Minimizers for a Class of Variational Problems with Polyconvex Integrand

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    Type
    Article
    Authors
    Awi, Romeo cc
    Sedjro, Marc
    Date
    2019-07-29
    Preprint Posting Date
    2017-02-05
    Online Publication Date
    2019-07-29
    Print Publication Date
    2020-08
    Permanent link to this record
    http://hdl.handle.net/10754/626531
    
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    Abstract
    We 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.
    Citation
    Awi, 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
    Sponsors
    Most 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.
    Publisher
    Springer Nature
    Journal
    Acta Applicandae Mathematicae
    DOI
    10.1007/s10440-019-00282-0
    arXiv
    1702.01471
    Additional Links
    http://link.springer.com/10.1007/s10440-019-00282-0
    ae974a485f413a2113503eed53cd6c53
    10.1007/s10440-019-00282-0
    Scopus Count
    Collections
    Publications Acknowledging KAUST Support; Computer, Electrical and Mathematical Science and Engineering (CEMSE) Division

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