Eikonal vs. Brownian: Regularity for the solution of an equation with gradient constraint.

Abstract: Two controllers are in charge of steering a spaceship in some domain Omega. The first controller wants to spend as much time as possible exploring Omega while the second wants to get out of it as quickly as possible. The first controller determines minute by minute whether the ship is moving by a Brownian motion or with constant speed, in which case it is the second controller who chooses the direction. Under these instructions, determining the optimal strategies for each player leads us to solve the equation $\min (-\Delta u, |Du|) = 1$ which has several interesting characteristics. Among them is the presence of a free boundary which separates the regions where a Poisson or an Eikonal equation is satisfied. In a recent collaboration with Edgard Pimentel (PUC-Rio) we showed that the solutions are Lipschitz continuous and that $|Du|$ is continuous, even though the gradient is discontinuous in numerous examples. This problem is a simplification of interesting models in financial mathematics related with the optimal strategy for the payment of dividends from multiple insurances.

analysis of PDEs

Audience: researchers in the topic

Rio de Janeiro webinar on analysis and partial differential equations

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