A1 Journal article (refereed)
On the heterogeneous distortion inequality (2021)

Kangasniemi, I., & Onninen, J. (2021). On the heterogeneous distortion inequality. Mathematische Annalen, Early online. https://doi.org/10.1007/s00208-021-02315-2

JYU authors or editors

Publication details

All authors or editors: Kangasniemi, Ilmari; Onninen, Jani

Journal or series: Mathematische Annalen

ISSN: 0025-5831

eISSN: 1432-1807

Publication year: 2021

Publication date: 29/11/2021

Volume: Early online

Publisher: Springer

Publication country: Germany

Publication language: English

DOI: https://doi.org/10.1007/s00208-021-02315-2

Publication open access: Not open

Publication channel open access:

Web address of parallel published publication (pre-print): https://arxiv.org/abs/2102.03471


We study Sobolev mappings f∈Wloc1,n(Rn,Rn), n≥2, that satisfy the heterogeneous distortion inequality |Df(x)|n≤KJf(x)+σn(x)|f(x)|n for almost every x∈Rn. Here K∈[1,∞) is a constant and σ≥0 is a function in Llocn(Rn). Although we recover the class of K-quasiregular mappings when σ≡0, the theory of arbitrary solutions is significantly more complicated, partly due to the unavailability of a robust degree theory for non-quasiregular solutions. Nonetheless, we obtain a Liouville-type theorem and the sharp Hölder continuity estimate for all solutions, provided that σ∈Ln−ε(Rn)∩Ln+ε(Rn) for some ε>0. This gives an affirmative answer to a question of Astala, Iwaniec and Martin.

Keywords: complex analysis; partial differential equations; differential geometry; inequalities (mathematics)

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Last updated on 2022-19-08 at 18:56