Serrin-type rigidity for the torsion problem in (ℝ^n, δ, |x|^α dx)

Determine whether, in the weighted Euclidean space (ℝ^n, δ, |x|^α dx) with α ≠ 0 and weighted Laplacian Δ_f = Δ − ∇f·∇ where f(x) = −α log|x|, the existence of a solution u to the overdetermined torsion problem Δ_f u = −1 in a bounded domain Ω with boundary conditions u = 0 and ∂_ν u = −c on ∂Ω forces Ω to be a Euclidean ball and u to be radially symmetric, thereby establishing a Serrin-type rigidity result in this weighted setting.

Background

Example 1 in the paper considers the weighted Euclidean space with power-law density e^{-f} = |x|^α. The authors explain that compatibility between curvature conditions and continuity of the weight is problematic (requiring α < 0 for nonnegativity of the Bakry–Émery tensor versus α ≥ 0 for continuity at the origin), placing this case outside the hypotheses of their main compact-result framework.

They explicitly point out that, for this widely studied weighted model, establishing whether the overdetermined problem characterizes balls (and radial solutions) remains unresolved and particularly challenging.