## Kuran’s Theorem

It is well-known that a harmonic function ${u:\Omega\rightarrow\mathbb{R}}$ satisfies the mean value property

$\displaystyle u(x)=\dfrac{1}{\left|B(x,r)\right|}\int_{B(x,r)}u(y)dy=\dfrac{1}{\left|\partial B(x,r)\right|}\int_{\partial B(x,r)}u(y)dS(y),$

for all balls ${B(x,r)}$ and spheres ${\partial B(x,r)}$ contained in ${\Omega}$. (We abuse notation and use ${\left|\cdot\right|}$ to denote both the ${n}$-dimensional Lebesgue measure and the ${(n-1)}$-dimensional surface measure.) This mean value property actually characterizes harmonic functions. A locally integrable function ${u}$ which satisfies the mean value property is harmonic; this is a form of Weyl’s lemma. Given this result, we could say that the mean value property over balls/spheres completely characterizes harmonic functions.

What if, instead, we know that a connected, open set (a domain) ${\Omega}$ has the property that every function ${u}$ which is harmonic in ${\Omega}$ satisfies the volume mean value property

$\displaystyle u(x)=\dfrac{1}{\left|\Omega\right|}\int_{\Omega}u(y)dy,\indent\forall x\in\Omega$

Is ${\Omega}$ necessarily an open ball of some radius ${r}$? After limited results were proved by B. Epstein [2] and M.M. Schiffer [3], \”{U}. Kuran provided a very elegant answer in the affirmative to this question for bounded domains [5]. As shown in [4], the hypothesis that ${\Omega}$ is connected is unnecessary, so we will trivially deviate from Kuran’s original result by dispensing with it.

Theorem 1 (Kuran) Let ${\Omega\subset\mathbb{R}^{n}}$ (${n\geq 2}$) be a bounded, open set. Suppose there is a point ${x_{0}\in\Omega}$ such that for every function harmonic in and integrable over ${\Omega}$,

$\displaystyle u(x_{0})=\dfrac{1}{\left|\Omega\right|}\int_{\Omega}u(x)dx$

Then ${\Omega}$ is an open ball (disk) centered at ${x_{0}}$.

Proof: The idea of the proof is pretty simple. Since ${\Omega}$ has compact closure, there exists a point ${x_{1}\in\partial\Omega}$ satisfying the equality

$\displaystyle r:=\left|x_{1}-x_{0}\right|=\inf_{x\in\Omega^{c}}\left|x-x_{0}\right|$

So ${B:=B(x_{0},r)\subset\Omega}$. If we can show that ${\Omega\subset\overline{B}}$, then since ${\text{int}(\overline{B})=B}$, it follows that ${\Omega=B}$. Assume the contrary: ${\Omega\setminus\overline{B}\neq\emptyset}$ and therefore has positive measure, being open.

Now suppose there is a function ${u}$ which vanishes at ${x_{0}}$ in ${\Omega}$ and is harmonic on the domain. Then it follows from the hypotheses and the ordinary mean value property applied to the ball ${B}$ that

$\displaystyle 0=u(x_{0})=\int_{\Omega}u(x)dx=\int_{\Omega\setminus B}u(x)dx+\underbrace{\int_{B}u(x)dx}_{=0}=\int_{\Omega\setminus B}u(x)dx$

From the last expression, we see that a nonnegative function ${u}$ cannot be strictly positive on a subset of ${\Omega\setminus B}$ of positive measure. This last observation suggests that by a sensible choice for ${u}$, we can arrive at a contradiction.

The Poisson kernel suggests that we consider the expression

$\displaystyle K(x):=\dfrac{\left|x-x_{0}\right|^{2}-r^{2}}{r^{2-n}\left|x-x_{1}\right|^{n}},$

which equals ${-1}$ at ${x_{0}}$. Since we can add a constant term without changing harmonicity, we are led to consider the function

$\displaystyle u(x):=\dfrac{\left|x-x_{0}\right|^{2}}{r^{2-n}\left|x-x_{1}\right|^{n}}+1$

Lemma 2 The function ${u}$ defined above vanishes at ${x_{0}}$, is harmonic on ${\mathbb{R}^{n}\setminus\left\{x_{1}\right\}}$, is integrable on ${\Omega}$, and is minorized by ${1}$ on ${\Omega\setminus B}$.

Proof: Computing partial derivatives (it’s a good exercise in calculus to work it out yourself), we see that

$\displaystyle \begin{array}{lcl} \displaystyle u_{x_{i}x_{i}}(x)&=&\displaystyle\dfrac{2}{r^{2-n}\left|x-x_{1}\right|^{n}}-\dfrac{4n(x_{i}-x_{0,i})(x_{i}-x_{1,i})}{r^{2-n}\left|x-x_{1}\right|^{n+2}}-\dfrac{n\left(\left|x-x_{0}\right|^{2}-r^{2}\right)}{r^{2-n}\left|x-x_{1}\right|^{n+2}}+\dfrac{n(n+2)\left(\left|x-x_{0}\right|^{2}-r^{2}\right)\left(x_{i}-x_{1,i}\right)^{2}}{r^{2-n}\left|x-x_{1}\right|^{n+4}} \end{array}$

whence

$\displaystyle \begin{array}{lcl}\displaystyle\Delta u(x)&=&\displaystyle\dfrac{2n}{r^{2-n}\left|x-x_{1}\right|^{n}}-\dfrac{4n(x-x_{0})\cdot(x-x_{1})}{r^{2-n}\left|x-x_{1}\right|^{n+2}}+\dfrac{2n\left(\left|x-x_{0}\right|^{2}-r^{2}\right)}{r^{2-n}\left|x-x_{1}\right|^{n+2}}\\[2 em] \displaystyle&=&\displaystyle\dfrac{2n\left|x-x_{1}\right|^{2}-4n(x-x_{0})\cdot(x-x_{1})+2n(\left|x-x_{0}\right|^{2}-r^{2})}{r^{2-n}\left|x-x_{1}\right|^{n+2}}\\ [2 em]\displaystyle&=&\displaystyle\dfrac{2n\left|(x-x_{1})-(x-x_{0})\right|^{2}-2nr^{2}}{r^{2-n}\left|x-x_{1}\right|^{n+2}}\\ [2 em]\displaystyle&=&\displaystyle0\end{array}$

It is evident that ${u\geq 1}$ on ${\Omega\setminus B}$. To see that ${u\in L^{1}(\Omega)}$, observe that ${\left|K(x)\right|\lesssim\left|x-x_{1}\right|^{1-n}}$ near the singularity ${x_{1}}$. $\Box$

We conclude that

$\displaystyle 0=\int_{\Omega}u(x)dx\geq\int_{\Omega\setminus B}1dx>0,$

which is a contradiction. This completes the proof. $\Box$

We conclude by mentioning some related results for the benefit of the interested reader. In a cutely titled paper [1], D. Aharonov, M.M. Schiffer, and L. Zalcman prove a related result for a class of spaces which the authors call potatoes. In [6], N. Suzuki and N.A. Watson prove the analogous inverse mean value theorem for solutions of the heat equation. Lastly, we mention that Kuran’s theorem has application to approximation theory. M. Goldstein, W. Haussman, and L. Rogge use Kuran’s theorem to prove existence and uniqueness of the best harmonic approximant in ${L^{1}}$ norm to a subharmonic function [4].

1. D. Aharonov, M.M. Schiffer, and L. Zalcman, Potato Kugel, Israel Jour. of Math. 40 (1981), 331-339.
2. B. Epstein, On the Mean-Value Property of Harmonic Functions, Proc. Amer. Math. Soc. 13 (1962), 830.
3. B. Epstein and M.M. Schiffer, On the Mean-Value Property of Harmonic Functions, J. Analyse Math. 14 (1965), 109-111.
4. M. Goldstein, W. Haussman, and L. Rogge, On the Mean Value Property of Harmonic Functions and Best Harmonic $L^{1}$-Approximation, Trans. Amer. Math. Soc. 305 (1988), 505-515.
5. U. Kuran, On the Mean-Value Property of Harmonic Functions, Bull. Lon. Math. Soc. (1972), 311-312.
6. N. Suzuki and N.A. Watson, A Characterization of Heat Balls by a Mean Value Property for Temperatures, Proc. Amer. Math. Soc. 129 (2001), 2709-2713.