Differential equation en $H^{1}(\Pi)$

Question

Good day, Please suggest me texts to study the existence and uniqueness of the solution in $H^{1}(\pi)$ of equation $x''+ wx=f$, with f in square-integrable functions $L^{2}(\pi)$.

$w>0$ and $\pi$ is el torus, the domain.

Answer 1

I'm not sure of a text, but you can work directly with the problem on $[0,2\pi]$. The equation $x''+wx=0$ with $x(0)=x(2\pi)$ and $x'(0)=x'(2\pi)$ has non-trivial solutions iff $w=n^2$ for some $n=0,1,2,3,\cdots$. For $w=0$, the solutions are constants. For $w=n^2$, where $n =1,2,3,\cdots$, the independent solutions are $e^{\pm inx}$, and $x''+wx=f$ is solvable for $x$ iff $(f,e^{\pm inx})=0$. For $w$ equal to any other real or complex value, the equation $x''+wx=f$ is uniquely solvable for $x\in H^2$ with periodic conditions, given $f\in L^2$.