Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 12 / 2 / iv / Solution

Take $\Omega =(0,\pi )$, $a_{11}=1$, $q=1$, $\psi =0$ and $f(x)=\sin x$. The coefficients are bounded and the operator is uniformly elliptic. If a weak solution $u\in H_0^1(0,\pi )$ existed, testing its weak formulation with $\phi =\sin x\in H_0^1(0,\pi )$ would giveBy integration by parts for $H_0^1$ functions, the first integral on the left is ${\int }_0^{\pi }u\sin x$, so the left side is zero. The right side equals $-\pi /2$, a contradiction. ThusThe forcing is not orthogonal to the homogeneous zero-boundary eigenfunction $\sin x$. This is the obstruction described by the Fredholm alternative for an elliptic Dirichlet problem, proved here directly without assuming that theorem.