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.
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
SN - 0944-2669
VL - 57
JO - Calculus of Variations and Partial Differential Equations
JF - Calculus of Variations and Partial Differential Equations
IS - 1
M1 - 13
ER -