If the dirichlet boundary condition is being applied, what does it tell us?
That's a pretty vague question!
Dirichlet condition specifies (or, "tells us") the values the solution to a (possibly, partial) differential equation must have on the boundary - as opposed to, for instance, the first or second derivative on the boundary. Alternative answer: the fact that the analyst is using Dirichlet condition "tells us" that those values are sufficient to determine the function throughout the domain (the interior which is enclosed by the boundary). That's often the case but it's also not uncommon that the derivative(s) are also required for a unique solution.
If one of those answers is not what you're looking for please explain your question further.
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