In some math books, instead of finding a strongly differentiable solution to some PDEs, they find weak solutions (weakly differentiable). How can we say that the weak solution satisfies that PDE? This is an example: Consider the eigenvalue problem \begin{equation}\tag{1} \begin{split} \Delta u+m(x)u=\lambda u; \Omega\\ \frac{\partial u}{\partial\eta}+\beta(x)u=0;\partial \Omega \end{split} \end{equation}
where $\lambda$ is a parameter, $m(x)\in L^\infty(\Omega)$, $\beta(x)\in L^\infty(\partial\Omega)$, $\frac{\partial u}{\partial\eta}$ is the outward normal derivative, and $\Omega$ is a bounded domain with $\partial\Omega$ smooth boundary.
This is how the principal eigenvalue $\lambda_1$ of the above problem is defined (and consequently the eigenfunction $\psi$),
\begin{equation}\tag{2} \lambda_1=\max_{\psi\in W^{1,2}(\Omega), \psi\neq 0}\big[\frac{-\int_\Omega|\nabla\psi|^2dx+\int_\Omega m(x)\psi^2dx-\int_{\partial\Omega}\beta(x)\psi^2ds}{\int_\Omega\psi^2dx}\big] \end{equation}
My concern is that, even if we have a function $\psi$ that satisfies (2), it may not twice differentiable (strongly) so how can we guarantee that it is a solution to (1)
