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

Let W ì Bbb RⁿOmega subset {Bbb R}^n be a smooth domain and let u ? C⁰(W).u in C^0(Omega ). A classical result of potential theory states that¶¶-ò_{Sr([`(x)])</sub> u(x)ds(x)=u([`(x)])-kern-5mmintlimits _{S_{r}(bar x)} u(x)dsigma (x)=u(bar x)¶¶for every [`(x)] ? Wbar xin Omega and r > 0r>0 if and only if¶¶Du=0 in W.Delta u=0 hbox { in } Omega.¶¶Here -ò<sub>Sr([`(x)])} u(x)ds(x)-kern-5mmintlimits _{S_{r}(bar x)} u(x)dsigma (x) denotes the average of u on the sphere S_r([`(x)])S_r(bar x) of center [`(x)]bar x and radius r. Our main result, which is a "localized" version of the above result, states:¶¶Theorem. Let u ? W^2,1(W)uin W^{2,1}(Omega ) and let x ? Wxin Omega be a Lebesgue point of DuDelta u such that¶¶-ò_Sr([`(x)]) u d s- a = o(r²)-kern-5mmintlimits _{S_{r}(bar x)} u d sigma - alpha =o(r^2)¶¶for some a ? Bbb Ralpha in Bbb R and all sufficiently small r > 0.r>0. Then¶¶Du(x)=0.Delta u(x)=0.