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Title: On established and new semiconvexities in the calculus of variations
Author: Kabisch, Sandra
ISNI:       0000 0004 5991 9476
Awarding Body: University of Surrey
Current Institution: University of Surrey
Date of Award: 2016
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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
Supervisor: Bevan, Jonathan J. Sponsor: University of Surrey
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available