Let $\Omega \subset \mathbb{R}^n$ be a compact smooth hypersurface. Suppose $\varphi \in C_c^\infty(0,T; H^1(\Omega))$ is a $H^1(\Omega)$-valued test function (it vanishes at so $t=0$ \varphi(t) \in H^1(\Omega)$for each$t$and$t= T$)\varphi(0) = \varphi(T)= 0$), and $f \in C^1(0,T C^1([0,T] \times \Omega)$. Let $w \in L^2(0,T;H^1(\Omega)$ L^2(0,T;H^1(\Omega))$with weak time derivative$w' \in L^2(0,T;H^{-1}(\Omega)$L^2(0,T;H^{-1}(\Omega))$ (i.e. it satisfies $\int_0^T w(t)\varphi'(t) = -\int_0^T w'(t)\varphi(t)$ for all such $\varphi$.)\varphi \in C_c^\infty(0,T;H^1(\Omega))$.) How do I show that $$\int_0^T \langle w', f\varphi \rangle_{H^{-1}, H^1} = \int_0^T \langle fw', \varphi \rangle_{H^{-1}, H^1}$$ where$\langle f,u \rangle_{H^{-1}, H^1} := f(u)$for$f \in H^{-1}$and$u \in H^1.$I tried writing the pairing as a functional and used RRT but to no avail. Appreciate any help.. (My question in MSE received no attention, so I posted it here. http://math.stackexchange.com/questions/260897/weak-derivative-and-continuous-functions-functionals-distributions) Edit Maybe I am asking the wrong question. Because what does$fw'$as a functional mean? Perhaps it's defined to satisfy$fw'(\varphi) = w'(f\varphi)$? If so, is this an appropriate thing to do? 2 added 204 characters in body Suppose$\varphi \in C_c^\infty(0,T; H^1(\Omega))$is a$H^1(\Omega)$-valued test function (it vanishes at$t=0$and$t= T$), and$f \in C^1(0,T \times \Omega)$. Let$w \in L^2(0,T;H^1(\Omega)$with weak time derivative$w' \in L^2(0,T;H^{-1}(\Omega)$(i.e. it satisfies$\int_0^T w(t)\varphi'(t) = -\int_0^T w'(t)\varphi(t)$for all such$\varphi$.) How do I show that $$\int_0^T \langle w', f\varphi \rangle_{H^{-1}, H^1} = \int_0^T \langle fw', \varphi \rangle_{H^{-1}, H^1}$$ I tried writing the pairing as a functional and used RRT but to no avail. Appreciate any help.. (My question in MSE received no attention, so I posted it here. http://math.stackexchange.com/questions/260897/weak-derivative-and-continuous-functions-functionals-distributions) Edit Maybe I am asking the wrong question. Because what does$fw'$as a functional mean? Perhaps it's defined to satisfy$fw'(\varphi) = w'(f\varphi)$? If so, is this an appropriate thing to do? 1 # weak derivative and continuous function Suppose$\varphi \in C_c^\infty(0,T; H^1(\Omega))$is a$H^1(\Omega)$-valued test function (it vanishes at$t=0$and$t= T$), and$f \in C^1(0,T \times \Omega)$. Let$w \in L^2(0,T;H^1(\Omega)$with weak time derivative$w' \in L^2(0,T;H^{-1}(\Omega)$(i.e. it satisfies$\int_0^T w(t)\varphi'(t) = -\int_0^T w'(t)\varphi(t)$for all such$\varphi\$.)
How do I show that $$\int_0^T \langle w', f\varphi \rangle_{H^{-1}, H^1} = \int_0^T \langle fw', \varphi \rangle_{H^{-1}, H^1}$$