A deterministic‐control‐based approach to fully nonlinear parabolic and elliptic equations

Abstract

We show that a broad class of fully nonlinear, second-order parabolic or elliptic PDEs can be realized as the Hamilton-Jacobi-Bellman equations of deterministic two-person games. More precisely: given the PDE, we identify a deterministic, discrete-time, two-person game whose value function converges in the continuous-time limit to the viscosity solution of the desired equation. Our game is, roughly speaking, a deterministic analogue of the stochastic representation recently introduced by Cheridito, Soner, Touzi, and Victoir. In the parabolic setting with no u-dependence, it amounts to a semidiscrete numerical scheme whose timestep is a min-max. Our result is interesting, because the usual control-based interpretations of second-order PDEs involve stochastic rather than deterministic control.

Very singular diffusion equations: second and fourth order problems

“…This is actually proved for the anisotropic Allen-Cahn equation by Giga et al [61] when the driving force term is spatially homogeneous but in arbitrary dimensions. It is interesting to study recent differential game approximation by Kohn and Serfaty [79,80] of a solution although approximation scheme for singular interfacial energy is not yet given. For example it is interesting to give a differential game interpretation for crystalline curvature flow.…”

Pde Approach to the Problem of Online Prediction With Expert Advice: A Construction of Potential-Based Strategies

“…From the results of [1,2,6] and the above calculations we expect that v is a viscosity subsolution of (7), (8). Fortunately, we need not care about the correctness of this conclusion, which is not evident in the present context.…”

Game theoretical methods in PDEs

“…In the case p = 1 the game is naturally related to the mean curvature flow [8] and functions of least gradient. Other extensions include the obstacle problems [15], finite difference schemes [2], equations with right hand side f = 0, mixed boundary data [1,3] and parabolic equations [9,14].…”