Uniqueness of lower semicontinuous viscosity solutions for the minimum time problem

We obtain the uniqueness of lower semicontinuous (LSC) viscosity solutions of the transformed minimum time problem assuming that they converge to zero on a `reachable' part of the target in appropriate direction. We present a counter-example which shows that the uniqueness does not hold without this convergence assumption. It was shown by Soravia that the uniqueness of LSC viscosity solutions having a `subsolution property' on the target holds. In order to verify this subsolution property, we show that the dynamic programming principle (DPP) holds inside for any LSC viscosity solutions. In order to obtain the DPP, we prepare appropriate approximate PDEs derived through Barles' inf-convolution and its variant.