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We determine Martin boundaries of product domains for elliptic equations in skew product form via Widder type uniqueness theorems. We show that the fiber of the Martin boundary at infinity of the base space degenerates into one point if any nonnegative solution to the Dirichlet problem for a corresponding parabolic equation with zero initial and boundary data is identically zero.

Uniqueness theorems for parabolic equations and Martin boundaries for elliptic equations in skew product form