Partial Differential Equations/Elliptic equations

Proposition (existence and uniqueness of weak solutions of a simple elliptic equation):

Let $I=[a,b]$ be a real interval and $A\in L^{\infty }(I)$ s.t. there exists $\alpha >0$ s.t. for almost every $x\in I$ we have $A(x)\geq \alpha$. The equation

${\begin{cases}-\partial _{x}(A(x)u(x))=f(x)&x\in I\\u(x)=0&x=a,b\end{cases}}$

admits a unique weak solution $u\in H_{0}^{1}(I)$.