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In weakly formulated PDE (or even ODE), we seem to be interested in solutions that satisfy or take desired values at some boundary points of a domain we are interested in. For example, Dirichlet boundary conditions impose the function to take zero or certain value on the boundary of the domain, and so are other types like periodic boundary conditions and Neumann boundary conditions, Robin boundary conditions, etc.,

My question is, can we also impose condition like, not only the function staisfy one of the above mentioned conditions, but also a new condition that function should take a so and so values at so and so points (finite in number) that lie in the interior of the domain.

Writing down as equation, let $P = \{p_i\}$ be a finite set of points in the interior of the domain, and condition is that $f(p_i) = d_i$, $d_i\in\mathbb{R}$ $d_i$ are the desired values the function should take at the points $p_i$

I want to know the questions in the title of this post, for the above mentioned conditions. The above pointwise conditions make sense only for weakly formulated PDE, so I am not interested in strong PDE.

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    $\begingroup$ One can also see these points $\{p_i\}$ as boundary points of the domain $\Omega\setminus \{p_i\}$. (Of course, they may or may not satisfy a given regularity property for the boundary) $\endgroup$ – Pietro Majer Nov 1 '17 at 9:29
  • $\begingroup$ @PietroMajer : Thats why I have edited the title, to remove the word "boundary". But after seeing your comment, I think the term 'boundary' is still apt. $\endgroup$ – Rajesh D Nov 1 '17 at 9:39
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Such interpolatory conditions are better treated by solving your PDE $Af=\sum_ic_i\delta_{p_i}$ as distributions in the whole domain, with the $c_i$s to be determined so that $f(p_i)=d_i$. For elliptic $A$ this is exactly what splines (in the variational sense) do, typically for $A=\partial_x^2$ (linear interpolation) or $A=\partial_x^4$ (cubic splines), in one dimension, or $A=\Delta^2$ in two dimensions (thin plate splines).

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    $\begingroup$ The problem with your formulation is, we cannot solve it, even numerically, for arbitrary number of points, in an arbitrary dimensional space . I hope I am wrong. $\endgroup$ – Rajesh D Jul 26 '18 at 5:50
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    $\begingroup$ You're right, not every dimension of space is possible for every operator. For example $A=\Delta$ is possible only in one dimension. The question is whether you need $Au=0$ in the whole domain, or outside the interpolatory points only. In the first case, it is possible to add as many interpolatory conditions as the dimension of the function space of solutions $u$ to $Au=0$ + boundary conditions. For $A=\Delta$ with Neumann boundary condition, this is $1$. $\endgroup$ – Jean Duchon Jul 27 '18 at 14:25

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