# 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?

Let’s consider a domain ${O = (-1, 1) \times (0,1)}$, and an equation

$\displaystyle u - u_{x} - 2x u_{y} = 0, \hbox{ on } O, \ u = 1, \hbox{ on } \partial O. \ \ \ \ \ (1)$

Its counter part of exit problem is the following. With underlying process

$\displaystyle X_{t} = x+ t, \ Y_{t} = y + 2xt +t^{2},$

and exit time ${\tau(x,y) = \inf\{t>0: (X_{t}, Y_{t}) \notin O\}}$ the value functions is defined as

$\displaystyle v(x,y) = e^{-\tau(x,y)}.$

For this simple exit problem, a straightforward computation leads to an explicit value ${v}$: With a decomposition of ${O = O_{1} \cup O_{2} \cup O_{3}}$ by

$\displaystyle O_{1} = \{y>x^{2}\}, \ O_{2} = \{y0\}, \ \hbox{ and } O_{3} = \{y

the value function can be expressed explicitly as

$\displaystyle v = e^{x - \sqrt{1 + x^{2} - y}} \hbox{ on } O_{1}; \ e^{x-1} \hbox{ on } O_{2}; \ e^{x+ \sqrt{x^{2} - y}} \hbox{ on } O_{3}.$

Indeed, one can check that ${v}$ is a generalized viscosity solution of (1). However, it is discontinuous inside ${O}$, in particular at any point on the set ${\partial O_{1} \cap \partial O_{3} \cap O}$.