## Mathematical Sciences Faculty Presentations

#### Title

A Nonlinear Differential Inequality and Counterexamples for Holder Continuous Almost Complex Structures

Presentation

12-1-2009

Analysis Seminar

Invited

#### Abstract

We consider a second order quasilinear partial differential inequality 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 df/dzbar=|f|^alpha, 0<\alpha<1, and f(0) not =0, there is also a lower bound for sup|f| on the unit disk. For each alpha, we construct a manifold with an alpha-Holder continuous almost complex structure where the Kobayashi-Royden pseudonorm is not upper semicontinuous, generalizing an example of Ivashkovich, Pinchuk, and Rosay.

#### Disciplines

Analysis | Mathematics

COinS