TY - JOUR

T1 - On the heterogeneous distortion inequality

AU - Kangasniemi, Ilmari

AU - Onninen, Jani

N1 - Funding Information:
J. Onninen was supported by the NSF grant DMS-1700274.
Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2022/12

Y1 - 2022/12

N2 - 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)|nfor 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.

AB - 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)|nfor 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.

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U2 - 10.1007/s00208-021-02315-2

DO - 10.1007/s00208-021-02315-2

M3 - Article

AN - SCOPUS:85120075288

SN - 0025-5831

VL - 384

SP - 1275

EP - 1308

JO - Mathematische Annalen

JF - Mathematische Annalen

IS - 3-4

ER -