A Hopf-Lax formula for the level-set equation and applications to PDE-constrained shape optimisation

Level-sets are a flexible method to describe geometries and their changes according to a speed field. This can be used in a wide variety of applications. We will present a Hopf-Lax formula that can be used to represent the solution of the level-set equation as well as the described geometries directly. This formula is a generalisation of existing results to the case of speed fields without a uniform, positive lower bound. The corresponding equation is of Hamilton-Jacobi type with a non-convex Hamiltonian. Our representation formula can be used both for theoretical and numerical purposes. In the latter case, the Fast Marching Method can be applied, leading to very efficient and robust numerical calculations of the geometry evolutions. We will also apply the level-set framework to an illustrative problem in PDE-constrained shape optimisation, and present numerical results.