Some nonlinear differential inequalities and an application to H\"older continuous almost complex structures

TL;DR

This paper establishes lower bounds for solutions to certain nonlinear differential inequalities and demonstrates an application to complex structures where the Kobayashi-Royden pseudonorm lacks upper semicontinuity.

Contribution

It introduces new lower bound results for nonlinear PDE inequalities and constructs examples of almost complex structures with non-semicontinuous pseudonorms.

Findings

Lower bounds for solutions to specific quasilinear PDE inequalities.

Existence of complex functions with prescribed differential properties.

Construction of manifolds with non-semicontinuous pseudonorms.

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 $f(0)\ne0$, there is also a lower bound for $\sup|f|$ on the unit disk. For each $\alpha$, we construct a manifold with an $\alpha$-H\"older continuous almost complex structure where the Kobayashi-Royden pseudonorm is not upper semicontinuous.