A Lavrentiev phenomenon in the neo-Hookean model

TL;DR

This paper demonstrates a Lavrentiev gap in the neo-Hookean energy model for 3D nonlinear elasticity, showing that certain boundary conditions lead to a strict energy gap due to a dipole singularity.

Contribution

It provides the first explicit example of a Lavrentiev phenomenon in the neo-Hookean model, highlighting the impact of singularities on energy minimization.

Findings

Existence of a Lavrentiev gap for the neo-Hookean energy in 3D.

Construction of boundary data causing the energy gap.

Identification of a dipole-type singularity as the mechanism.

Abstract

We exhibit a Lavrentiev gap phenomenon for the neo-Hookean energy in three-dimensional nonlinear elasticity. More precisely, we construct boundary data for which the infimum of the neo-Hookean energy over deformations satisfying a natural regularity and invertibility condition is strictly larger than the infimum over the weak $H^1$-closure of that class. The mechanism underlying the gap is a deformation with a dipole-type singularity.