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- Title
ON THE EXISTENCE OF MINIMIZERS FOR THE NEO-HOOKEAN ENERGY IN THE AXISYMMETRIC SETTING.
- Authors
Henao, Duvan; Rodiac, Rémy
- Abstract
Let Ω be a smooth bounded axisymmetric set in R3. In this paper we investigate the existence of minimizers of the so-called neo-Hookean energy among a class of axisymmetric maps. Due to the appearance of a critical exponent in the energy we must face a problem of lack of compactness. Indeed as shown by an example of Conti-De Lellis in [12,Section 6], a phenomenon of concentration of energy can occur preventing the strong convergence in W1,2(Ω,R3) of a minimizing sequence along with the equi-integrability of the cofactors of that sequence. We prove that this phenomenon can only take place on the axis of symmetry of the domain. Thus if we consider domains that do not contain the axis of symmetry then minimizers do exist. We also provide a partial description of the lack of compactness in terms of Cartesian currents. Then we study the case where Ω is not necessarily axisymmetric but the boundary data is affine. In that case if we do not allow cavitation (nor in the interior neither at the boundary) then the affine extension is the unique minimizer, that is, quadratic polyconvex energies are W1,2-quasiconvex in our admissible space. At last, in the case of an axisymmetric domain not containing its symmetry axis, we obtain for the first time the existence of weak solutions of the energy-momentum equations for 3D neo-Hookean materials.
- Subjects
CONVEX functions; ENERGY momentum relationship; AXIAL flow; TOPOLOGICAL degree; CARTESIAN coordinates
- Publication
Discrete & Continuous Dynamical Systems: Series A, 2018, Vol 38, Issue 9, p4509
- ISSN
1078-0947
- Publication type
Article
- DOI
10.3934/dcds.2018197