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- W2283667114 abstract "Let ({mathbb{X} subset mathbb {R}^n}) be a bounded Lipschitz domain and consider the energy functional $${{mathbb F}_{sigma_2}}[u; mathbb{X}] := int_mathbb{X} {mathbf F}(nabla u) , dx,$$over the space of admissible maps $${{mathcal {A}_varphi}(mathbb{X}) :={u in W^{1,4}(mathbb{X}, {mathbb{R}^n}) : {rm det}, nabla u > 0, {rm for}, {mathcal {L}^n}{rm -a.e. in}, mathbb{X}, u|_{partial mathbb{X}} =varphi }},$$where the integrand ({{mathbf F} colon mathbb M_{ntimes n} to mathbb{R}}) is quasiconvex and sufficiently regular. Here our attention is paid to the prototypical case when ({{mathbf F}(xi):=frac{1}{2} sigma_2(xi)+Phi(detxi)}). The aim of this paper is to discuss the question of multiplicity versus uniqueness for extremals and strong local minimizers of ({mathbb F_{sigma_2}}) and the relation it bares to the domain topology. In contrast, for constructing explicitly and directly solutions to the system of Euler–Lagrange equations associated to ({{mathbb F}_{sigma_2}}), we use a topological class of maps referred to as generalised twists and relate the problem to extremising an associated energy on the compact Lie group ({mathbf {SO}(n)}). The main result is a surprising discrepancy between even and odd dimensions. In even dimensions the latter system of equations admits infinitely many smooth solutions amongst such maps whereas in odd dimensions this number reduces to one." @default.
- W2283667114 created "2016-06-24" @default.
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- W2283667114 date "2016-02-12" @default.
- W2283667114 modified "2023-10-17" @default.
- W2283667114 title "$${sigma_2}$$ σ 2 -Energy as a polyconvex functional on a space of self-maps of annuli in the multi-dimensional calculus of variations" @default.
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- W2283667114 doi "https://doi.org/10.1007/s00030-016-0372-3" @default.
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