Some nonlinear differential inequalities and an application to Hölder continuous almost complex structures

Abstract

We consider some second order quasilinear partial differential inequalities for real-valued functions on the unit ball and find conditions under which there is a lower bound for the supremum of nonnegative solutions that do not vanish at the origin. As a consequence, for complex-valued functions $f(z)$ satisfying $\partial f/\partial \bar{z}=|f{|}^{\alpha }$, $0<\alpha <1$, and $f(0)\ne 0$, there is also a lower bound for $\sup |f|$ on the unit disk. For each $\alpha$, we construct a manifold with an $\alpha$-Hölder continuous almost complex structure where the Kobayashi–Royden pseudonorm is not upper semicontinuous.