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. 



On a conjecture of Lions for the wave equation
You will find your answer in this article:


<http://www.sciencedirect.com/science/article/pii/0022247X89902059>


