Classification of Nonnegative g −Harmonic Functions in Half-Spaces

Abstract

In this paper we present a short proof of the following classification Theorem for g −harmonic functions in half-spaces. Assume that u is a nonnegative solution to Δgu = 0 in {xn > 0} that continuously vanishes on the flat boundary {xn = 0}. Then, modulo normalization, u(x) = xn in {xn ≥ 0}. Our proof depends on a recent quantitative version of the Hopf-Oleinik Lemma proven by the authors in Braga and Moreira (Adv. Math.334, 184–242, 2018). Moreover, in this paper, we show how to adapt the proofs in the literature to extend Carleson Estimate, Boundary Harnack Inequality and Schwartz Reflection Principle to the context of nonnegative g −harmonic functions. These results are also ingredients for the proof of the main result.