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?

Let’s consider a domain , and an equation

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

and exit time the value functions is defined as

For this simple exit problem, a straightforward computation leads to an explicit value : With a decomposition of by

the value function can be expressed explicitly as

Indeed, one can check that is a generalized viscosity solution of (1). However, it is discontinuous inside , in particular at any point on the set .

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