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Some nonlinear differential inequalities and an application to Hölder continuous almost complex structures. (English) Zbl 1213.35409
Summary: 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 \(f(0)\neq 0\), 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.

MSC:
35R45 Partial differential inequalities and systems of partial differential inequalities
32F45 Invariant metrics and pseudodistances in several complex variables
32Q60 Almost complex manifolds
32Q65 Pseudoholomorphic curves
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35P15 Estimates of eigenvalues in context of PDEs
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