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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?

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    @Nick The context of this question is someone's first encounter with the Dirichlet problem. In this context, it is usual to consider continuous boundary data. Besides, continuity allows one to easily treat rough boundaries (snowflakes etc) where there is no obvious choice of measure to integrate against.2012-09-21

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The answer is no: there is no implicit assumption that $u$ is harmonic on a superset of the closure. In fact, the boundary data $f$ may well be some kind of continuous but nowhere differentiable function like Weierstrass's. That would obviously preclude any possibility of harmonic extension of $u$ to a larger domain.

Added I think I see the source of confusion. You are right that the function $u=\partial P/\partial \theta $ has radial limit zero at every point of the boundary. But this does not mean that $u$ is continuous on the closed disk. Indeed, it is not even bounded on the disk. The Dirichlet problem (with continuous boundary data) requires continuous extension to the boundary, not just the existence of radial limits.

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    I see, finally. Thanks very much for @timur and @LVK!2012-09-21