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 $A\in L^{\mathrm{\infty }}(I)$ s.t. there exists $\alpha >0$ s.t. for almost every $x\in I$ we have $A(x)\ge \alpha$. The equation

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

admits a unique weak solution $u\in H_0^1(I)$.