In this talk we will discuss a model for the accretive growth of an elastic solid. The reference configuration of the body is accreted in its normal direction, with space- and deformation- dependent accretion rate. The time-dependent reference configuration is identified via the level sets of the unique viscosity solution of a suitable generalized eikonal equation. As a corollary of a regularity result for level sets of solutions to the stationary Hamilton-Jakobi equation, we will show John regularity of the level sets of the aforementioned generalized eikonal equation.
After proving the global-in-time well-posedness of the quasistatic equilibrium under prescribed growth, we will prove the existence of a local-in-time solution for the coupled equilibrium-growth problem, where both mechanical displacement and time-evolving set are unknown.
The talk is based on joint work with K. Nik, U. Stefanelli, and G. Tomassetti.