I will present a new method to prove the solvability of a class of nonlinear elliptic PDEs, namely Hamilton-Jacobi-Bellman equations, in the whole space and with merely measurable and unbounded coefficients. Such equations arise mainly in stochastic optimal control, but also in asymptotic problems in PDEs (homogenisation, long-time behaviour, ...).

The method relies on duality and optimisation in abstract Banach spaces, together with thorough analysis of diffusion operators. The main idea is to derive the solution of the PDE from the dual variable of a suitably chosen optimisation problem. I will also discuss how the method can be generalised to tackle other PDEs.

References:

[1] H. Kouhkouh, A viscous ergodic problem with unbounded and measurable ingredients. Part 1: HJB equation. (SIAM J. Contr. Opt., 2024)

[2] H. Kouhkouh, A viscous ergodic problem with unbounded and measurable ingredients. Part 2: Mean-Field Games. (arXiv:2311.04616)

[3] H. Kouhkouh, The viscous eikonal equation in the whole space with nonsmooth right-hand side and stable drift. (In preparation)