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Heinz‐type inequality and bi‐Lipschitz continuity for quasiconformal mappings satisfying inhomogeneous biharmonic equations
Author(s) -
Zhong Deguang,
Zhou Yu,
Yuan Wenjun
Publication year - 2019
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.5611
Subject(s) - biharmonic equation , mathematics , lipschitz continuity , type (biology) , mathematical analysis , class (philosophy) , boundary (topology) , pure mathematics , boundary value problem , ecology , artificial intelligence , computer science , biology
Let Hom + ( T ) be the class of all sense‐preserving homeomorphic self‐mappings of T = { z = x + i y ∈ C : | z | = 1 } . The aim of this paper is twofold. First, we obtain Heinz‐type inequality for ( K , K ′ )‐quasiconformal mappings satisfying inhomogeneous biharmonic equation Δ(Δ ω ) = g in unit disk D with associated boundary value conditions Δ ω | T = φ ∈ C ( T ) and ω | T = f ∗ ∈ Hom+ ( T ) . Second, we establish biLipschitz continuity for ( K , K ′ )‐quasiconformal mappings satisfying aforementioned inhomogeneous biharmonic equation when K ′ , | | φ | | ∞ : = sup z ∈ D | φ ( z ) | and | | g | | ∞ : = sup z ∈ D | g ( z ) | are small enough.