# Positive subharmonic functions with constant integral blowing up at boundary

Say, we’re given smooth functions $$f_n$$, $$n=1,2,3,…$$ defined on a smooth bounded domain $$Omegasubsetmathbb{R}^d$$ satisfying

1. $$Delta f_nge 0$$ (subharmonic)
2. $$f_nge 0$$
3. $$int_Omega f_n=I>0$$ for all $$ninmathbb{N}$$
4. $${f_n}_{|partialOmega}=n$$

Then, say $$Bsubsetsubset Omega$$. Can we conclude that $$int_B f_nto 0$$?

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.

MathOverflow Asked by Fozz on February 7, 2021

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 $$1-1/nleq|x|leq 1$$. 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.

Answered by Alexandre Eremenko on February 7, 2021

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