Abstract
In this work our main objective is to establish various (high frequency-) uniqueness criteria. Initially, we consider \(p\)-Dirichlet type functionals on a suitable class of measure preserving maps \(u: B\subset \mathbb{R}^{2} \to \mathbb{R}^{2}\), \(B\) being the unit disk, and subject to suitable boundary conditions. In the second part we focus on a very similar situations only exchanging the previous functionals by a suitable class of \(p\)-growing polyconvex functionals and allowing the maps to be arbitrary.
In both cases a particular emphasis is laid on high pressure situations, where only uniqueness for a subclass, containing solely of variations with high enough Fourier-modes, can be obtained.
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Notes
For the notation regarding Fourier-series consult the end of §.2 below.
Recall, that we call situations, where the energies remain finite either incompressible elastic, if the considered admissible maps must be measure-preserving, otherwise we call it compressible elastic. Moreover, we call a model (fully) non-linear elastic, if the considered integrand \(f\), ignoring any other dependencies, satisfies \(f(\xi )=+\infty \) for any \(\xi \in \mathbb{R}^{n\times n}\) s.t. \(\det \xi \le 0\) and \(f(\xi )\rightarrow +\infty \) if \(\det \xi \rightarrow 0^{+}\) or \(\det \xi \rightarrow +\infty \). Notice, that in the latter model some of the considered energies might be infinite.
See, [22, Prop A.1] with \(\sigma =0\).
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Acknowledgements
The author is in deep dept to Jonathan J. Bevan, Bin Cheng, and Ali Taheri for vital discussions and suggestions, the fantastic people with the Department of Mathematics at the University of Surrey and the Engineering & Physical Sciences Research Council (EPRSC), which generously funded this work. We thank the referee for valuable comments and remarks.
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Dengler, M. (High Frequency-) Uniqueness Criteria for \(p\)-Growth Functionals in in- and Compressible Elasticity. J Elast 154, 607–618 (2023). https://doi.org/10.1007/s10659-023-09996-7
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DOI: https://doi.org/10.1007/s10659-023-09996-7