Consider a sequence $(f_n)$ of functions in the flat torus $T^d$ converging Lebesgue-a.e. to a limit function $f$.

Assume that:

1) $|f_n|(x)\leq 1$ for every $n,x$

2) $\Delta f_n\geq -1$ in the sense of distributions for every $n$

Can we deduce that $$ \lim_{n\to\infty}\int|\nabla f_n|^2 d\mathcal L^d=\int|\nabla f|^2 d\mathcal L^d $$ ?

Notice that $(1)+(2)$ grant that $\int|\nabla f_n|^2 d\mathcal L^d=-\int f_nd\Delta f_n\leq \mathcal L^d(T^d)<\infty$ for every $n$ so that the sequence is bounded in $W^{1,2}$ and thus, given the a.e. convergence, also weakly converging.

The question is then if it is strongly converging in $W^{1,2}$.

In fact, I'm interested in this problem in a much more general setting, the functions being defined on a converging sequence of metric measure spaces with Ricci curvature bounded from below, but after some thinking I realized that I don't know the answer not even in this simplified setting.

Some comments. The assumption on the Laplacians ensures that they are measures with uniformly bounded mass, and the uniform bound on the $W^{1,2}$-norms of the functions grant that $|\Delta f_n(E)|$ can be bounded from above in terms of the capacity of $E$ only.

Therefore to conclude it would be sufficient to show that for every $\epsilon$ there exists $\delta,N$ such that for $n>N$ the set $\{|f_n-f|>\delta\}$ has capacity smaller than $\epsilon$ (here it matters to pick the upper semicontinuous representatives of the functions).

I've searched in the literature and found results which are very close to this one, but not really sufficient for my purposes, see for instance the book `Growth Theory of Subharmonic Functions' by Vladimir Azarin.

Yet, I found no reference for the question I'm asking.

Any help would be appreciated, thanks in advance.