In Control of Distributed Singular Systems p 236, JL Lions makes the conjecture :

Let $\Omega$ be a domain in $\mathbb{R}^n$, $Q = \Omega \times ]0,T[$ and consider

$\phi'' - \triangle \phi = F$

$\phi(x,0) = \phi^0(x), \phi'(x,0) = \phi^{1}(x), \phi^0 \in H^1(\Omega), \phi^1 \in L^2(\Omega)$

$\frac{\partial \phi}{\partial \nu} = 0 $ on $\Sigma$

where $F \in L^1(0,T;L^2(\Omega))$.

Then, the conjecture is that $\phi \in H^1(\Sigma)$.

Since this book was written almost 30 years ago, I was wondering if there are any results about this conjecture. Thanks in advance for any insight.