# A subharmonic function in a ball

This question is related to this (A nonnegative harmonic function in a ball). Let $$B(a,r)$$ be an open ball in $$\mathbb{R}^m$$, ($$m\geq2$$). Is there a function $$u$$, $$u\not\equiv0$$, that is subharmonic on $$B(a,r)$$, harmonic on a neighborhood of $$0$$, and $$u(a)=0$$?

A function $$u$$ defined on a domain $$D$$ of $$\mathbb{R}^m$$ with $$m\geq 2$$ and with values in $$[-\infty,\infty)$$ is called subharmonic, if

1) $$u$$ is upper semi continuous, meaning that for each $$x\in D$$, $$\limsup_{y\to x}u(x)\leq u(x),$$ (limisup is taken from inside $$D$$)and

1. $$u(x)\leq \frac{1}{d_mr^m}\int_{B(x,r)}u(y)dy,$$ where $$d_mr^m$$ is the volume of the ball $$B(x,r)$$.

Let me explain what is the use of such construction. This way from a function that is subharmonic locally, we can construct a function that is subharmonic on a neighborhood of the infinity, including at infinity. This a well-know result that if $$u$$ is subharmonic on a ball $$B(a,r)$$, harmonic on a neighborhood of $$a$$ and $$u(a)=0$$, then the function defined by $$u^*(x^*)=\Big(\frac{r}{|x^*-a|}\Big)^{m-2}u(x)$$ for $$|x^*|\geq r$$ and $$0$$ at infinity, is subharmonic for $$|x^*|\geq r$$ (see Lester L. Helms, Potential Theory, pg 208)

• where is effort ? Please write what you are thinking
– MAS
Apr 3, 2021 at 18:17
• I got it! Thanks. Apr 4, 2021 at 0:52
• Please define "subharmonic". Apr 4, 2021 at 3:09
• Sorry! It is now done. Apr 4, 2021 at 22:09

Let $$v(x)=|x|$$ where $$|.|$$ designates the Euclidean norm in $$\mathbb{R}^m$$. This is a subharmonic function. We take $$0 and replace $$v$$ by its Poisson's integral on $$B(0,r')$$ and leave as $$v$$ on the rest of $$B(0,r)$$. Then the function we obtain, say $$v'$$, is harmonic on a neighborhood of $$0$$. Finally to make it $$0$$ at the origin we set $$u(x)=v'(x)-v'(0).$$ I think it works.