Consider functional $$I_a^b(u) = \int_a^b u(t) dt$$ defined on the Sobolev space $H_0^1([c,d])$, where $c<a<b<d$. As this functional is bounded, from Riesz representation theorem we know that there exists $\varphi \in H_0^1([c,d])$ such that $$\int_c^d \mathbb{1}_{[a,b]} u(t)dt = \int_c^d \varphi(t)u(t)dt + \int_c^d \varphi'(t)u'(t)dt.$$
Q: What is $\varphi$ in this particular example? I tried to start with $\varphi_{[a,b]} \equiv 1$ and then extend it somehow onto the rest of the domain, but I think that can't possibly be the case since then we would need to have $$\int_{[c,d]\setminus[a,b]} \varphi(t)u(t)dt +\varphi'(t)u'(t)dt = 0$$ and this looks like $\varphi$ would need to be zero for this to hold for every $u$.
