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

1 Answers

One Answer

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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