Regularities for PDE Solution

Question

I consider the following PDE: $$ -u''=f \quad \text{ in } \Omega = (0,1) \\ u(0) =u(1)=0.$$ Which can, by multiplication with a testfunction and partial integration, be rewritten as $$\int_\Omega u'v' dx = \int_\Omega fv dx.$$ Which we call weak formulation. As far, as I understand, the weak form reduces the demanded regularity from $u \in C^2(\Omega)$ to $u \in C^1(\Omega)$, yielding a larger solution space. But why do we start with $C^2(\Omega)$? Since there are functions $u$, where $u''$ exists, but $u'' \notin C(\Omega)$, meaning that the second derivative of the solution does not have to be continuous. Only, if $f$ is demanded to be continuous. But usually there is not much said about $f$.

Answer 1

You are not told much about $f$ because you are not studying a particular ODE (PDEs involve differentiation in more than one variable). Instead you are studying a family of ODEs where the function $f$ is allowed to vary over some domain. The more you know about that domain, the more powerful theorems you can prove about $u$. But this comes at a price of having more limited application for those theorems. So one has to strike a balance between power and width applicability of the results.

$f\in C(\Omega)$ is a good balance. Continuity is a pretty light requirement for most applications - particularly physical ones, and even when a discontinuous function is useful, it can generally be treated as a limit of continuous ones. But continuity has a lot of powerful implications on $u$.

So the introduction to a family of ODEs or PDEs is generally for continuous functions. But this is where the study of the subject starts, not where it stops. Expanding the set of functions $f$ to see what still can be proved, restricting the set to allow stronger theorems to be proved - these are things that mathematicians engage in all the time. But students are introduced to the field with the easy stuff.