[HG_AP] Rigidity and flexibility in polyconvex elastodynamics
ABSTRACT
We study rigidity and flexibility phenomena for weak solutions of nonlinear elastodynamics associated with smooth stored-energy functions. A central question in nonlinear elasticity is how structural conditions on the stored energy, such as polyconvexity and strong quasiconvexity, influence the existence, regularity, and uniqueness of weak solutions.
In this talk, I will first discuss the flexibility side of the problem. Using convex integration, we construct smooth, strongly polyconvex stored-energy functions for which the associated elastodynamic systems admit infinitely many Lipschitz weak solutions for a broad class of initial and boundary data. These solutions are highly oscillatory and nowhere continuously differentiable in space-time. The construction is based on a differential-inclusion framework involving $\mathcal{T}_N$-configurations.
I will then contrast these flexibility results with known rigidity results based on weak–strong uniqueness for dissipative measure-valued solutions. Interestingly, the stored-energy functions arising in our construction satisfy the structural assumptions used in such rigidity theories. Thus, rigidity and convex-integration flexibility can occur within the same class of constitutive laws, depending crucially on the notion of solution and the imposed admissibility and regularity conditions. I will conclude with some questions concerning possible regularity thresholds separating the rigid and flexible regimes.
