Weak Solutions and Test Functions

Multiply the equation by a smooth helper, integrate by parts, and the derivative moves off the unknown onto the helper. The price is a boundary bracket, but test functions are smooth helpers engineered to vanish near the boundary, so the bracket dies every time. So for every test function phi, the integral of u prime phi equals minus the integral of u phi prime, and that identity defines u prime when u has no classical derivative. Doing it twice moves two derivatives, and the two minus signs cancel. What remains is the weak formulation: an equation that asks less of u yet means the same whenever u is smooth.

A Sobolev space collects the functions whose weak derivatives stay square-integrable: it keeps u plus its weak derivatives in L2. The point of these spaces is completeness, so limits of approximating sequences stay inside and computations converge to real members. In one dimension every H1 function has a continuous representative, so tents are welcome but jumps are not. The step function is in L2 yet has no L2 weak derivative, since differentiating its jump gives a point mass.

For smooth functions the weak machinery changes nothing: x squared has weak second derivative 2, matching the classical one. The absolute value shows the limit: its first weak derivative jumps from negative 1 to 1, and differentiating that jump concentrates a point mass at the kink. So it has no weak second derivative in L2. When moving derivatives, always check which side of the kink you stand on.