question about formulation of partial differential equations Given a bounded open set $\Omega$ and some differential operator, for example the laplacian operator $\Delta$, one PDE formulation is to find $u$
$$
\Delta u = f
$$ 
in $\Omega$, and $u = g$ on $\partial \Omega$. In this formulation, do we mean the PDE should be satisfied only in $\Omega$? or do we also want the PDE to hold on the boundary, that is
$$\Delta u(x) = f(x)$$ for $x \in \partial \Omega$? 
 A: We want $\Delta u = f$ to be satisfied only in $U$ because derivative operators are not well-defined in the boundary $\partial U$. 
Derivative operators in PDEs are defined using limit processes; for example, the usual Frechét derivative is the (unique) linear operator $A$ such that
$$\lim_{h \to 0} \frac{\|u(x+h)-u(x)-Ah\|}{\|h\|} = 0$$
The problem with taking this limit at the boundary is that we don't have any information whatsoever about $u(x+h)$ for $x+h \not\in \overline{U}$, thus the limit from that direction is not defined. Similar problems arise with the laplacian. 
One could ask (assuming that $f$ is actually defined outside $U$):

Could we extend $u(x,t)$ by forcing it to satisfy $\Delta u = f$ outside $U$ , say, for another open set $V \supset \overline{U}$, while still having $u=g$ in $\partial U$? 

Then you lose uniqueness; there is an infinite number of smooth functions that satisfy the equation in $V$ while still satisfying the "boundary" condition in $\partial U$.
And, if you impose conditions over $\partial V$, well, it's no different than considering the problem in $V$ in the first place.
