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The primary objective of the present work is the establishment of a "Boundary Harnack Principle" for harmonic functions which vanish on part of the boundary of a Lipschitz domain in Rn and a corresponding result for solutions of the heat equation. The statements of these properties arise naturally from a not-so-familar formulation of the standard Harnack principle, a formulation which is suggested by a necessary condition for the uniqueness statement of the principle of positive singularities.

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