Harmonicity of functions satisfying a weak form of the mean value property

Abstract.

Let $\Omega \subset {\Bbb R}^n$ be a smooth domain and let $u \in C^0(\Omega ).$ A classical result of potential theory states that¶¶$-\kern-5mm\int\limits _{S_{r}(\bar x)} u(x)d\sigma (x)=u(\bar x)$¶¶for every $\bar x\in \Omega$ and $r>0$ if and only if¶¶$\Delta u=0 \hbox { in } \Omega.$¶¶Here $-\kern-5mm\int\limits _{S_{r}(\bar x)} u(x)d\sigma (x)$ denotes the average of u on the sphere $S_r(\bar x)$ of center $\bar x$ and radius r. Our main result, which is a “localized” version of the above result, states:¶¶Theorem. Let $u\in W^{2,1}(\Omega )$ and let $x\in \Omega$ be a Lebesgue point of $\Delta u$ such that¶¶$-\kern-5mm\int\limits _{S_{r}(\bar x)} u d \sigma - \alpha =o(r^2)$¶¶for some $\alpha \in \Bbb R$ and all sufficiently small $r>0.$ Then¶¶$\Delta u(x)=0.$