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.

My guess is that since $w$ has a weak derivative in time (real line), then it must be differentiable for almost all $t\geq 0$ (which is not necessarily true in higher dimension, for example differentiation with respect to space $x$). So $\int_\Omega w(t,x)dx$ must also be differentiable for almost all $t\geq 0$ and so we can use some kind of dominated convergence theorem.


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.