Given a family of measurable functions with , shall we have measurable? If each is continuous, then is lower semicontinuous, so is measurable. However, it may not be true in general.
This is an example of Dirichlet problem whose unique solution only meets its boundary in the generalized sense. In other words, there is no solution in classical Dirichlet sense. It is taken from Example 7.8 of [Crandall, Ishii, Lion 1992].
Recalling that, strong comparison principle is able to identify a unique viscosity solution in . In this below, we have an example having its interior being discontinuous. Is there any theory to justify its uniqueness? Continue reading
In this below, we will discuss some connections on the several representations on the generator of Levy process. Continue reading
Ito’s formula provides probabilistic representation of PDE with Cauchy terminal data backward in time. By simple change of variable in time, it also provides a useful probabilistic representation of PDE with Cauchy initial data, which recovers Duhamel’s formula. Continue reading