Harmonicity of functions satisfying a weak form of the mean value property (Q5943210)

Let \(\Omega\subset\mathbb{R}^n\), be a smooth domain, \(U\in C^0 (\Omega)\), \(S_r(\overline x)\) -- the sphere of center \(\overline x\) and radius \(r\) and \(W^{k,p}(\Omega)\) -- the Sobolev space of functions whose weak derivatives of order \(\alpha\), \(|\alpha|\leq k\), belong to \(L^p_{\text{loc}}(\Omega)\). It is known that a classical result of potential theory states that \(\oint_{S_r(\overline x)}u(x) d \delta(x)= u(\overline x)\) for every \(\overline x\in\Omega\) and \(r>0\) if and only if \(\Delta u=0\) in \(\Omega\). Here \(\oint_{S_r (\overline x)}u(x) d\delta(x)\) denotes the average of \(u\) on the sphere \(S_r(\overline x)\). The main result of this paper is a ``localized'' version of the above result, i.e. the following Theorem. Let \(u\in W^{2,1} (\Omega)\) and let \(x\in\Omega\) be a Lebesgue point of \(\Delta u\) such that \(\oint_{S_r(x)} u d\delta-\alpha= o(r^2)\) for some \(\alpha\in \mathbb{R}\) and all sufficiently small \(r>0\). Then \(\Delta u(x)=0\).