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On established and new semiconvexities in the calculus of variations.

Kabisch, Sandra (2016) On established and new semiconvexities in the calculus of variations. Doctoral thesis, University of Surrey.

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After introducing the topics that will be covered in this work we review important concepts from the calculus of variations in elasticity theory. Subsequently the following three topics are discussed: The first originates from the work of Post and Sivaloganathan [\emph{Proceedings of the Royal Society of Edinburgh, Section: A Mathematics}, 127(03):595--614, 1997] in the form of two scenarios involving the twisting of the outer boundary of an annulus $A$ around the inner. It seeks minimisers of $∫_A \frac{1}{2}|∇u|^2 \d x$ among deformations $u$ with the constraint $\det ∇u ≥0$ a.e.~as well as of $∫_A \frac{1}{2}|∇u|^2 + h(\det ∇u) \d x$ in which $h$ penalises volume compression so that $\det ∇u > 0$ a.e.~is imposed on minimisers. In the former case we find infinitely many explicit solutions for which $\det ∇u = 0$ holds on a region around the inner boundary of $A$. In the latter we expand on known results by showing similar growth properties of the solutions compared to the previous case while contrasting that $\det ∇u>0$ holds everywhere. In the second we introduce a new semiconvexity called $n$-polyconvexity that unifies poly- and rank-one convexity in the sense that for $f:ℝ^{d×D}→\bar{ℝ}$ we have that $n$-polyconvexity is equivalent to polyconvexity for $n=\min\{d,D\}=:d∧D$ and equivalent to rank-one convexity for $n=1$. For $d,D≥3$ we gain previously unknown semiconvexities in hierarchical order ($2$-polyconvexity, \dots, $(d∧D-1)$-polyconvexity, weakest to strongest). We further define functions which are `$n$-polyaffine at $F$' and find that they are not necessarily polyaffine for $n<d∧D$ (unlike rank-one affine functions). As one of the main results we obtain that $1$-polyconvex (i.e.~rank-one convex) functions $f:ℝ^{d×D} →ℝ$ are the pointwise supremum of $1$-polyaffine functions at $F$ for every $F∈ℝ^{d×D}$. In addition and among other things, we discuss envelopes, generalised $T_k$ configurations and relations to quasiconvexity. The third involves a generalisation of the theory of abstract convexity which allows one to include cases like $1$-polyconvex functions as the pointwise supremum of $1$-polyconvex functions at $F$ for every $F$, while this is not possible within the classical theory. We review the most important results of the classical theory and present results on generalised hull operators, subgradients, conjugations and Legendre-Fenchel transforms for our new theory. In particular we obtain an operator that is reminiscent of the rank-one convexification process via lamination steps for a function. Moreover, we show that directional convexity is a special case of the generalised abstract convexity theory. Finally, we conclude each topic, pointing out possible directions of further research.

Item Type: Thesis (Doctoral)
Subjects : Mathematics, Calculus of Variations
Divisions : Theses
Authors :
Date : 30 September 2016
Funders : University of Surrey (University Research Scholarship)
Contributors :
ContributionNameEmailORCID, Jonathan
Depositing User : Sandra Käbisch
Date Deposited : 27 Oct 2016 09:17
Last Modified : 31 Oct 2017 18:41

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