I'm studying the following differential equation:
-(pu')'+qu={\lambda}wu
where $p,q,w$ are $[a,b]{\longrightarrow}{\mathbb R}, \lambda\in{\mathbb C}$ is a spectral parameter (not important), $p,w>0$ and $\frac{1}{p},q,w{\in}L_{\it loc}^1[a,b]$ (that is to say, locally absolutely integrable).
Now, $u$ and u' aren't necessarily supposed to be continuously differentiable: it is only the case that $u$ and pu' must be absolutely continuous*. On the other hand, pu', which two authors call the 'pseudo-derivative', does have to be differentiable. I have searched the internet to learn more about pseudoderivatives but the word doesn't bring up anything on any of the usual mathematical sites.
I would really appreciate it if somebody were able to answer these two questions that are on my mind.
- If $u$ is not supposed to be differentiable, how can we write pu', which clearly has a differentiated $u$ in it?
- If you know more about this type of differential equation, does ensuring pu' is differentiable tend to be a heavy constraint?
Sorry if these seem naive questions. I am writing an essay which does not really deal with pseudo-derivatives in a big way and mentions these constraints in passing. I am trying not to plunge headlong into the theory of weak solutions which would seriously derail my work.
Thank you if anybody is able to advise.
- Initially I did not make clear that pu' was required to be absolutely continuous. This is relevant to one of the points raised in the answer below.