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

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

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

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