On established and new semiconvexities in the calculus of variations.
Sandra Käbisch · Surrey Research Insight Open Access (The University of Surrey) · 2016
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