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} = \{y<x^{2}, x>0\}, \ \hbox{ and } O_{3} = \{y<x^{2}, x<0 \}

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}.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s