The boundary Harnack principle for non-divergence form elliptic operators
Bass, Richard F.
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If L is a uniformly elliptic operator in non–divergence form, the boundary Harnack principle for the ratio of positive L–harmonic functions holds in Hölder domains of order [alpha] if [alpha] > 1/2. A counterexample shows that 1/2 is sharp. For Hölder domains of order [alpha] with [alpha is an element of the set] (0, 1], the boundary Harnack principle holds provided the domain also satisfies a strong uniform regularity condition.