Let $\Omega$ be a bounded and regular open subset $\Omega$ of $\mathbb{R}^N$ and $u:[0,\infty)\times \Omega\to \mathbb{R}$ be a smooth function (for example a smooth solution to a PDE). Thus the function $w=\min(0,u)$ has weak time derivative given by $w_t=u_t.1_{\{u \leq 0 \}}$. Does the following "weak differentiation under the integral sign" holds? $$\frac{d}{dt}\int_\Omega w(t,x)dx=\int_\Omega \frac{\partial}{\partial t}w(t,x)dx,$$ for almost all $t\geq 0$.

We know that differentiation under the integral sign holds for $u$ because it is smooth. But I am wondering if it also holds for a function like $w=\min(0,u)$ which only has a weak derivative. If possible, I would like to ask for a reference addressing such a result.

  • $\begingroup$ Hint: try integrating both sides from $0$ to $t$ and applying Fubini's theorem. $\endgroup$ – Nate Eldredge Aug 12 at 22:20
  • $\begingroup$ I think you need to assume a bit more for the classical result (about $u$), otherwise examples such as $u(t,x)=x\sin (t/x)$ on $x\in\Omega=(0,1)$ become problematic. $\endgroup$ – Christian Remling Aug 12 at 22:26

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.