| Abstract: |
| Let $\Omega \subset \mathbb R^n$ be a bounded domain with Lipschitz boundary and let $\Omega_+\subset \Omega$ be a sub-domain with the exterior ball property. We establish existence and uniqueness of viscosity solutions for a class of transmission problems governed by elliptic and eikonal type equations in $\Omega_+$ and $\Omega_- := \Omega\setminus \Omega_+$ respectively. The main motivation is the Hamilton-Jacobi equation that results of the following optimal control problem: The goal is to minimize the expected time a particle takes from some initial position $x\in \Omega$ until it exits $\Omega$ for the first time. The controller is allowed to choose at each moment the direction that the particle takes whenever this is in the region $\Omega_-$, being the speed equal to one. Over $\Omega_+$ the particle performs instead a Brownian motion. |
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