Fundamental lemma of calculus of variations for higher order-derivatives

Question

Consider an expression of the form $$0=\int_a^b f_0(t) \varphi(t) + f_1(t)\varphi'(t) + \ldots + f_n(t) \varphi^{(n)}(t) \, \mathrm{d}t, \qquad \forall \varphi \in C_c^\infty(a,b)$$ and $f_0, \ldots, f_n \in L^2(a,b)$. Can I conclude that $$f_i \in W^{i,2}(a,b)$$ where $W^{i,2}(a,b)$ denotes the set of all $f \in L^2(a,b)$ whose weak-derivatives up to order $i$ exist and are again in $L^2(a,b)$? Especially in the case $n=1$, this is the definition of the weak derivative. I would be very grateful for hints.

Answer 1

No, this equation only implies $$ \sum_{j=0}^n (-1)^j f_j^{(j)}=0 $$ almost everywhere, where $f_j^{(j)}$ denotes the $j$-th order weak derivative of $f_j$.

A counterexample to your claim would be $(a,b)=(-1,+1)$, $$ f_0 = 0, \quad f_1 = \chi_{(0,1)}, \quad f_2 = -\max(x,0). $$ Here, $f_1 \in L^2 \setminus H^1$, $f_2 \in H^1 \setminus H^2$.