Minimization Problems for the Axisymmetric Neo-Hookean Energy

Explore minimization problems for the axisymmetric neo-Hookean energy in this 38-minute talk by Marco Barchiesi, presented at the Workshop on "Between Regularity and Defects: Variational and Geometrical Methods in Materials Science" held at the Erwin Schrödinger International Institute for Mathematics and Physics (ESI). Delve into the existence of minimizers for the 3D neo-Hookean energy in the critical case, where the Sobolev exponent p=2. Examine the phenomenon of energy concentration in the axisymmetric case, as demonstrated by Conti-De Lellis's example, which hinders strong convergence of minimizing sequences and equi-integrability of cofactors. Learn how this lack of compactness can be transformed into a regularity problem through a specific relaxed energy. Gain insights into the bounds provided for this relaxed energy and their implications for solving minimization problems in materials science.