= Solution
Fix a closed ball $\overline{B_R(a)}\subset\Omega$, and let $v$ solve the <Dirichlet problem> for the <Laplace equation> in this ball with boundary data $u$. Put $w=u-v$. The function $w$ is continuous, vanishes on the boundary, and inherits the restricted spherical mean identity because $v$ has the full <mean value property for harmonic functions>.
Suppose $M=\max_{\overline{B_R(a)}}w>0$. Its maximum set $E$ is a nonempty compact subset of the open ball. Choose $x\in E$ maximizing $|x-a|$. For every sufficiently small radius in the sequence attached to $x$,
$$
M=w(x)=\frac1{|\partial B_r|}\int_{\partial B_r(x)}w\leq M.
$$
Equality of the average with the maximum and <continuous function>[continuity] imply that the whole sphere belongs to $E$. Its point in the direction from $a$ through $x$ lies farther from $a$ than $x$ does; if $x=a$, any point on the sphere does. Both cases contradict the choice of $x$. Hence $w\leq0$, and applying the same argument to $-w$ gives $w=0$.
Thus $u=v$ on every relatively compact ball. It is consequently harmonic and smooth locally, proving the <local converse to the mean value property>.
Solved by gpt-5.6-sol high.
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