Discontinuous Solutions of the Hamilton--Jacobi Equation for Exit Time Problems

Abstract

In general, the value function associated with an exit time problem is a discontinuous function. We prove that the lower (upper) semicontinuous envelope of the value function is a supersolution (subsolution) of the Hamilton--Jacobi equation involving the proximal subdifferentials (superdifferentials) with subdifferential-type (superdifferential-type) mixed boundary condition. We also show that if the value function is upper semicontinuous, then it is the maximum subsolution of the Hamilton--Jacobi equation involving the proximal superdifferentials with the natural boundary condition, and if the value function is lower semicontinuous, then it is the minimum solution of the Hamilton--Jacobi equation involving the proximal subdifferentials with a natural boundary condition. Futhermore, if a compatibility condition is satisfied, then the value function is the unique lower semicontinuous solution of the Hamilton--Jacobi equation with a natural boundary condition and a subdifferential type boundary condition. Some conditions ensuring lower semicontinuity of the value functions are also given.