Some nonlinear differential inequalities and an application to Hölder continuous almost complex structures
Annales de l'I.H.P. Analyse non linéaire, Volume 28 (2011) no. 2, p. 149-157

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 $f\left(z\right)$ satisfying $\partial f/\partial \overline{z}={|f|}^{\alpha }$, $0<\alpha <1$, and $f\left(0\right)\ne 0$, there is also a lower bound for $\mathrm{sup}|f|$ on the unit disk. For each α, we construct a manifold with an α-Hölder continuous almost complex structure where the Kobayashi–Royden pseudonorm is not upper semicontinuous.

DOI : https://doi.org/10.1016/j.anihpc.2011.02.001
Classification:  35R45,  32F45,  32Q60,  32Q65,  35B05
Keywords: Differential inequality, Almost complex manifold
@article{AIHPC_2011__28_2_149_0,
author = {Coffman, Adam and Pan, Yifei},
title = {Some nonlinear differential inequalities and an application to H\"older continuous almost complex structures},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
publisher = {Elsevier},
volume = {28},
number = {2},
year = {2011},
pages = {149-157},
doi = {10.1016/j.anihpc.2011.02.001},
zbl = {1213.35409},
mrnumber = {2784067},
language = {en},
url = {http://www.numdam.org/item/AIHPC_2011__28_2_149_0}
}

Coffman, Adam; Pan, Yifei. Some nonlinear differential inequalities and an application to Hölder continuous almost complex structures. Annales de l'I.H.P. Analyse non linéaire, Volume 28 (2011) no. 2, pp. 149-157. doi : 10.1016/j.anihpc.2011.02.001. http://www.numdam.org/item/AIHPC_2011__28_2_149_0/

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