# 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

## Related Questions

### What are some good resources for mathematical translation?

7  Asked on January 5, 2022 by matt-noonan

### Inequality for hook numbers in Young diagrams

2  Asked on January 5, 2022 by igor-pak

### Most tensor subspaces of low dimension have rank-1 defining equations

0  Asked on January 5, 2022

### Prime ideals of formal power series ring that are above the same prime ideal

1  Asked on January 5, 2022 by a-c-biller

### Pull back of a Bounded form

0  Asked on January 5, 2022 by kamel

### Set-theoretic generation by circuit polynomials

0  Asked on January 5, 2022

### How to prove that $sum_{m=0}^{infty} { Gamma{(1+2m)/alpha}over Gamma(1/2+m)} { (-t^2/4)^{m}over m !} ge (alpha/2)^{3}exp(-t^{2}/4)$

0  Asked on January 3, 2022

### Clarification regarding the definition of absolute height of an algebraic number

0  Asked on January 3, 2022

### Appearance of proof relevance in “ordinary mathematics?”

2  Asked on January 3, 2022 by mcncm

### Reference to a Classical Regularity Theorem

1  Asked on January 3, 2022 by artful_dodger

### About Countable Dense Homogeneous spaces (CDH) and strongly locally homogeneous spaces

0  Asked on January 3, 2022 by gabriel-medina

### Is there an abstract logic that defines the mantle?

1  Asked on January 3, 2022

### Lorentzian vs Gaussian Fitting Functions

5  Asked on January 1, 2022 by jimmidyjoo

### Applications of mathematics in clinical setting

7  Asked on January 1, 2022

### Bounds on number of “non-metric” entries in matrices

1  Asked on January 1, 2022

### Eigenvalues of convolution matrices

0  Asked on January 1, 2022 by sina-baghal

### Given a large random matrix, how to prove that every large submatrix whose range contains a large ball?

0  Asked on January 1, 2022

### Does the $K^1$-group of a complete flag variety vanish?

0  Asked on January 1, 2022 by quin-appleby

### Does every special $C^*$-Frobenius algebra have a unit?

0  Asked on January 1, 2022 by quantumorange

### Motives under de-singularization

0  Asked on January 1, 2022 by user161697

### Ask a Question

Get help from others!