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].

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

\displaystyle  u + x u_{y} = 0, \hbox{ on } O, \ u = \phi(x,y) := y, \hbox{ on } \partial O. \ \ \ \ \ (1)

Its counterpart of exit problem is the following.

  1. With underlying process

    \displaystyle X_{t} = x, \ Y_{t} = y - xt,

  2. and its exit time {\tau = \tau(x,y) = \inf\{t>0: (X_{t}, Y_{t}) \notin O\}}.
  3. The value functions is defined as

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

For this simple exit problem, a straightforward computation leads to an explicit value

\displaystyle v(x, y) = e^{(1- y)/x} I_{(-\infty, 0)}(x).

One can check that {v\in C(O)} satisfies the viscosity solution property in {O}. However, it loses some boundary values, in particular on {\{(x,1): x \ge 0\} \subset \partial O}. It is proved in Example 7.8 of [CIL92] that it’s not possible to have a solution with Dirichlet boundary values in the classical sense. This is the motivation of generalized Dirichlet problem. By imposing the viscosity property at the boundary points losing its boundary value, one can justify the above value {v} as a unique solution in this new definition.

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