Say, we're given smooth functions $f\_n$, $n=1,2,3,...$ defined on a smooth bounded domain $\Omega\subset\mathbb{R}^d$ satisfying


1. $\Delta f\_n\ge 0$ (subharmonic)
2. $f\_n\ge 0$
3. $\int\_\Omega f\_n=I>0$ for all $n\in\mathbb{N}$
4. ${f\_n}\_{|\partial\Omega}=n$


Then, say $B\subset\subset \Omega$. Can we conclude that $\int\_B f\_n\to 0$?


When I visualize these functions, I suspect this might be true, but I can't come up with a proof nor a counterexample. Any help would be appreciated.



Positive subharmonic functions with constant integral blowing up at boundary
Let $\Omega$ be the unit ball, $B$ some smaller concentric ball, and
$u\_n(x)=1$ for $|x|\leq 1-1/n$ and $u\_n(x)=n(n-1)|x|+2n-n^2$ for $1-1/n\leq|x|\leq 1$.
Then your conditions 1,2,4 are satisfied exactly, and 3 is satisfied approximately
(integrals tend to a positive constant), so a slight modification will give you
constant integrals, if really needed.


