A discussion on the solvability of Dirichlet problem

Last Friday, I’ve had a pleasant discussion with Professor Lions (here) on my recent paper (here). PDF is also available (here).

Supremum of uncountably many measurable functions may not be measurable

Given a family of measurable functions ${f_{\alpha}(x)}$ with ${\alpha \in \Lambda}$, shall we have ${g(x) = \sup_{\alpha\in \Lambda} f_{\alpha}(x)}$ measurable? If each ${f_{\alpha}}$ is continuous, then ${g(x)}$ is lower semicontinuous, so is measurable. However, it may not be true in general.

A motivating example for a generalized Dirichlet problem

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].

Uniqueness of a generalized Dirichlet problem: An Example

Recalling that, strong comparison principle is able to identify a unique viscosity solution in ${C(O)}$. In this below, we have an example having its interior being discontinuous. Is there any theory to justify its uniqueness? Continue reading

A note on the generator of some Levy process

In this below, we will discuss some connections on the several representations on the generator of Levy process. Continue reading

A representation of a heat equation with initial value

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