In one dimensional boundary value problems, one considers the equation of type $$Df(x)=g(x);x\in[a,b]$$ with boundary conditions of the type $$f(a)α_1+f′(a)α_2=0\\f(b)β_1+f′(b)β_2=0$$
However, in 3D on closed surfaces, my book says for Poisson's equation either we can specify the value of the function (Dirichlet) or its normal derivative (Neumann) but not both, for it leads to over specification.
My question is, why exactly this happens? What's wrong with closed surfaces? Could we do it in 2D? Is every surface in 1D open in this sense?
Also, is this over specification case true only for Poisson's equation, or a whole class of equations?
I'm from physics background
The difference between open & closed surfaces is that the Neumann & Dirichlet problems have unique solutions only for the closed surfaces.
In the case of a Neumann problem on a closed surface, the normal derivative is specified, and the value of the function itself is the unique result (at least to within an arbitrary constant). If you tried to also specify Dirichlet conditions, they would not generally be consistent with that solution. A similar argument holds when you are solving the Dirichlet problem.
When the surfaces are open, the nonuniqueness allows both types of conditions to be specified without conflict.
The above comments apply equally to 2D curves or 3D surfaces. In 1D, there is no such thing as a normal derivative, so no Neumann problem to consider.
They also obviously apply to Laplace's equation (since that is a special case of Poisson's equation), but I don't know more generally about other PDE's.