Bounded RMVP implies harmonicity in ℝ^n for n ≥ 3

Determine whether, in Euclidean dimensions n ≥ 3, every continuous bounded real-valued function u on a domain Ω ⊂ ℝ^n that satisfies the Restricted Mean Value Property—meaning that for each x ∈ Ω there exists a sphere S_x centered at x with radius ρ(x) > 0 contained in Ω such that u(x) equals the average of u over S_x with respect to the surface volume measure—must be harmonic in Ω; equivalently, ascertain whether a counterexample exists in dimension n ≥ 3.

Background

The Restricted Mean Value Property (RMVP) asks whether a function that agrees with its spherical average on at least one sphere centered at each point is necessarily harmonic. In bounded domains, boundary behavior plays a crucial role: while counterexamples exist without boundary assumptions, those are unbounded.

Littlewood posed the classical ‘one-circle problem’ in dimension n = 2: if a function satisfying RMVP is assumed to be bounded, is it harmonic? Hansen and Nadirashvili showed this has a negative answer in the unit disk, constructing a bounded continuous function satisfying RMVP that is not harmonic. The analogous question for higher dimensions remains unresolved.

References

Littlewood asked (in [Li68]) for n = 2, whether the unboundedness is the only obstruction, more precisely, if the function satisfying the RMVP is also assumed to be bounded, then is it harmonic? This is the classical ‘one-circle problem’ and in [HN94], Hansen and Nadirashvili showed that the above problem has a negati2e answer, that is, there exists a continuous, bounded function on the unit disk in R which satisfies RMVP but is not harmonic. The problem is still open for n ≥ 3 however.

— Restricted Mean Value Property with non-tangential boundary behavior on Riemannian manifolds (2401.09293 - Dewan, 17 Jan 2024) in Section 1 (Introduction)