So I am studying for a qualifying examination and there was this problem from an old exam.
(a) Does the problem $\Delta u = 0$ in the unit square in the plane with u and and all of its partial derivatives satisfying periodic boundary conditions have a unique solutions? If not, characterize all solutions.
(b) For smooth f, does the problem $\Delta u = f$ in the unite square, with u and all of its partial derivatives satisfying periodic boundary conditions generally have a solution or are there conditions on f? Give a proof of your conclusion.
I feel like I can approach (a) to some degree. However, I am having a hard time feeling like I am characterizing $all$ solutions. ie, separation of variables leads to the set of ODE's $P''(x) + \lambda^2P(x) = 0$ and $Q''(y) + \lambda^2Q(y)= 0$ and the periodic boundary conditions lead to $P(0)Q(y)=P(1)Q(y)$ and $P(x)Q(0) = P(x)Q(1)$. To illustrate my confustion in the simplest matter take $\lambda = 0$. Then both $P,Q$ are linear. If at least one is constant than it satisfies the equation and boundary conditions. But if they are both linear, it is not clear to me how to show that a solution is, or is not possible. That is I get if $P(x) = ax + b, Q(y) = cy + d$ then Bndry Conditions implies $b(cy+d) =(a+b)(cy+d)$ and $d(ax+b)=(ax+b)(c+d)$
Moreover, I want to translate the bndry conditions to imply $P(0) = P(1), Q(0) = Q(1)$ but it seems like this is only true if $Q(y),P(x) \ne 0$ for some $0 \le x,y \le 1$ and it certainly could be.
So yeah, I'm unsure of a real affective approach
For part (b) things are even worse. I'm not sure what to require except it seems reasonable that we would want f to periodic on the boundary (at least the limit as it approaches the boundary). But besides that I don't know. The most general starting point I can think of would be the general solution:
$u(x) = \int_U \psi(x-y)f(y)dy - \int_{\partial U} g(y)\frac{\partial \psi(x-y)}{\partial b} - \psi(x-y)\frac{\partial u(y)}{\partial n}dS(y)$
Anywho, any suggestions would be much appreciated! Thanks for your time!
The issue here is, the problem seems to set you up to use certain results, but those theorems are irrelevant in the problem as actually stated.
So long as the boundary conditions lie in the appropriate trace space ($H^{-\frac{1}{2}}$ of the boundary), and $f$ lies in $H^{-1}$ of the unit square, you can solve the equation on the square in the weak sense; this is just the Lax-Milgram theorem, which you can look up in any textbook on PDEs.
However, this has nothing to do with periodicity, which I don't know if the problem is hinting at. If you did not transcribe the problems faithfully, this may matter (no shame if you did not or you are working from another language - this happens all the time in math, and while it causes confusion, people get over it quickly).
If you want a bounded periodic solution to Laplace's equation in the plane (which is to say, it solves the equation everywhere, not just in tiles of the unit square), then you can invoke the Liouville theorem, which says that bounded harmonic functions are constants only.
Similarly, if you want a bounded periodic function $u$ that solves $\Delta u = f$ everywhere with $f$ periodic, you need to have $\int_S f = 0$ where $S$ is the unit square, you see this by considering the integral $$ 0 = \int_{\partial S} u_\nu = \int_S \Delta u = \int_S f$$ where the first equality follows from the periodicity of $u$ and the remainder are integration by parts.