Take the Dirichlet Problem as example:
The difinition of Dirichlet Problem from wiki
Given a function $f$ that has values everywhere on the boundary of a region in $\mathbb{R}^n$, is there a unique continuous function $u$ twice continuously differentiable in the interior and continuous on the boundary, such that $u$ is harmonic in the interior and $u=f$ on the boundary?
Does 'harmonic in the interior' implicitly and necessarily means that 'harmonic on an open superset of the closed region'?
Should I interpret a PDE problem this way? Namely, a PAE problem is actually defined on the open super set of the problem region?