Does a green's function of a Laplacian, in one dimensions, with periodic boundary conditions, exist?

Question

solve $$y'' = \delta(x-0.25)$$ $$y(0) = y(1), y'(0) = y'(1)$$ Does this have a solution? I couldn't construct one with this boundary conditions and would like to see a solution if it exists.

Answer 1

$$y(x) = ax+b+c(x-t)H(x-t)$$

$$y' = a+cH(x-t)$$

$$y'' = c\delta(x-t)$$

$$y(0) = b$$ $$y(1) = a + b + c$$ $$y'(0) = a$$ $$y'(1) = a+c$$

$$a+c = 0$$ $c = 0$ $a = 0$

Hence such a periodic green's function of the Laplacian not possible.

Answer 2

You can ask for a solution of the boundary problem or for a Green kernel for the operator $L[y]=y''$, but not both.

As the question for the Green kernel is the more general one, consider the more general equation for it, that is, for $y(x)=G(a,x)$ consider $$ y''=δ(x−a) $$ for any fixed $a\in(0,1)$. This implies a jump of $1$ in $y'$ at $x=a$, while outside that point the derivative is constant. This means that you can not have the same derivative at both interval ends, as $y'(1)=1+y'(0)$.