In general, the value function associated with an exit time problem is a di
scontinuous function. We prove that the lower (upper) semicontinuous envelo
pe of the value function is a supersolution (subsolution) of the Hamilton J
acobi equation involving the proximal subdifferentials (superdifferentials)
with subdifferential-type (superdifferential-type) mixed boundary conditio
n. We also show that if the value function is upper semicontinuous, then it
is the maximum subsolution of the Hamilton Jacobi equation involving the p
roximal 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 sa
tis ed, then the value function is the unique lower semicontinuous solution
of the Hamilton Jacobi equation with a natural boundary condition and a su
bdifferential type boundary condition. Some conditions ensuring lower semic
ontinuity of the value functions are also given.