*Submitted Paper*

**Inserted:** 16 mar 2021

**Last Updated:** 16 mar 2021

**Year:** 2021

**Abstract:**

In a previous work, Henao & Rodiac (2018) proved an existence result for the neo-Hookean energy in 3D for the essentially 2D problem of axisymmetric domains away from the symmetry axis. In the relaxation approach for the neo-Hookean energy in 3D, minimizers are found in the space of maps that can be obtained as the weak $H^1$-limit of a sequence of diffeomorphisms; see Malý (1993). Among the singular maps contained in that $H^1$-weak closure, a particularly upsetting pathology, constructed upon the dipoles found in harmonic map theory, was presented by Conti & De Lellis (2003). Their example involves a cavitation-type discontinuity around which the orientation is reversed, the cavity created there being furthermore filled with material coming from other part of the body. Since physical minimizers should not exhibit such behaviour, this raises the regularity question of proving that minimizers have Sobolev inverses (since the inverses of the harmonic dipoles have jumps across the created surface). The first natural step is to address the problem in the axisymmetric setting, without the assumption that the domain is hollow and at a distance apart of the symmetry axis. Attacking that problem will presumably demand a thorough analysis of the fine properties of singular maps in the $H^1$-sequential weak closure, as well as an explicit characterization of the relaxed energy functional. In this paper we introduce an explicit energy functional (which coincides with the neo-Hookean energy for regular maps and expresses the cost of a singularity in terms of the jump and Cantor parts of the inverse) and an explicit admissible space (which contains all weak $H^1$-limits of regular maps) for which we can prove the existence of minimizers. Chances to succeed in establishing that minimizers do not have dipoles are higher in this explicit alternative variational problem than when working with the abstract relaxation approach in the abstract space of all weak limits.

Our candidate for relaxed energy has many similarities with the relaxed energy introduced by Bethuel-Brézis-Coron for a problem with lack of compactness in harmonic maps theory. The proof we present in this paper for the lower semicontinuity of the augmented energy functional, partly inspired by the prominent role played by conformality in that context, further develops the connection between the minimization of the 3D neo-Hookean energy with the problem of finding a minimizing smooth harmonic map from \(\mathbb{B}^3\) into \(\mathbb{S}^2\) with zero degree boundary data.

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