TY - JOUR

T1 - On Lipschitz continuity of solutions of hyperbolic Poisson’s equation

AU - Chen, Jiaolong

AU - Huang, Manzi

AU - Rasila, Antti

AU - Wang, Xiantao

N1 - Publisher Copyright:
© 2017, Springer-Verlag GmbH Germany, part of Springer Nature.
Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.

PY - 2018/2/1

Y1 - 2018/2/1

N2 - In this paper, we investigate solutions of the hyperbolic Poisson equation Δ hu(x) = ψ(x) , where ψ∈ L∞(Bn, Rn) and (Formula Presented.)Δhu(x)=(1-|x|2)2Δu(x)+2(n-2)(1-|x|2)∑i=1nxi∂u∂xi(x)is the hyperbolic Laplace operator in the n-dimensional space Rn for n≥ 2. We show that if n≥ 3 and u∈ C2(Bn, Rn) ∩ C(Bn¯ , Rn) is a solution to the hyperbolic Poisson equation, then it has the representation u= Ph[ ϕ] - Gh[ ψ] provided that u∣Sn-1=ϕ and ∫Bn(1-|x|2)n-1|ψ(x)|dτ(x)<∞. Here Ph and Gh denote Poisson and Green integrals with respect to Δ h, respectively. Furthermore, we prove that functions of the form u= Ph[ ϕ] - Gh[ ψ] are Lipschitz continuous.

AB - In this paper, we investigate solutions of the hyperbolic Poisson equation Δ hu(x) = ψ(x) , where ψ∈ L∞(Bn, Rn) and (Formula Presented.)Δhu(x)=(1-|x|2)2Δu(x)+2(n-2)(1-|x|2)∑i=1nxi∂u∂xi(x)is the hyperbolic Laplace operator in the n-dimensional space Rn for n≥ 2. We show that if n≥ 3 and u∈ C2(Bn, Rn) ∩ C(Bn¯ , Rn) is a solution to the hyperbolic Poisson equation, then it has the representation u= Ph[ ϕ] - Gh[ ψ] provided that u∣Sn-1=ϕ and ∫Bn(1-|x|2)n-1|ψ(x)|dτ(x)<∞. Here Ph and Gh denote Poisson and Green integrals with respect to Δ h, respectively. Furthermore, we prove that functions of the form u= Ph[ ϕ] - Gh[ ψ] are Lipschitz continuous.

KW - Primary: 31B05

KW - Secondary: 31C05

UR - http://www.scopus.com/inward/record.url?scp=85039057191&partnerID=8YFLogxK

U2 - 10.1007/s00526-017-1290-x

DO - 10.1007/s00526-017-1290-x

M3 - 文章

AN - SCOPUS:85039057191

VL - 57

JO - Calculus of Variations and Partial Differential Equations

JF - Calculus of Variations and Partial Differential Equations

SN - 0944-2669

IS - 1

M1 - 13

ER -