A Boundary Value Problem for Bihypermonogenic Functions in Clifford ...

Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 974714, 9 pages http://dx.doi.org/10.1155/2014/974714

Research Article A Boundary Value Problem for Bihypermonogenic Functions in Clifford Analysis Xiaoli Bian1 and Yuying Qiao2 1 2

School of Science, Tianjin University of Technology and Education, Tianjin 300222, China College of Mathematics and Information, Hebei Normal University, Shijiazhuang 050016, China

Correspondence should be addressed to Xiaoli Bian; [email protected] Received 8 May 2014; Revised 25 June 2014; Accepted 1 July 2014; Published 20 July 2014 Academic Editor: Guo-Cheng Wu Copyright © 2014 X. Bian and Y. Qiao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper deals with a nonlinear boundary value problem for bihypermonogenic functions in Clifford analysis. The integrals of quasi-Cauchy’s type and Plemelj formula for bihypermonogenic functions are firstly reviewed briefly. The nonlinear Riemmann boundary value problem for bihypermonogenic functions is discussed and the existence of solutions is obtained, which also indicates that the linear boundary value problem has a unique solution.

1. Introduction Clifford algebra is an associative and noncommutative algebraic structure that was set up at the beginning of the twentieth century. Clifford analysis is an important branch of modern analysis, which studies the functions defined in R𝑛+1 with the value in Clifford algebra space [1]. Clifford analysis possesses not only important theoretical value but also applicable value, which plays an important role in many fields, such as quantum mechanics, Maxwell equation, and YangMills field. Since 1987, Xu [2, 3], Wen [4], Huang [5, 6], Qiao [7–9], and so forth have done a lot of work on boundary value problems for monogenic functions and biregular functions in Clifford analysis. Eriksson and Leutwiler [10–12] introduced hypermonogenic functions in Clifford analysis, studied some of its properties, and discussed the integral representation for hypermonogenic functions. Qiao [9] investigated the boundary value problems of hypermonogenic functions. In recent years, Zhang and Du [13, 14] discussed Riemann boundary value problems and singular integral equations in Clifford analysis. Bian et al. [15] obtained the integral formulas and Plemelj formula for bihypermonogenic functions. Yang et al. [16] studied a kind of boundary value problem for hypermonogenic function vector. Zhang and Gürlebeck [17] studied Riemann boundary value problems in Clifford analysis.

In this paper, based on the integral formulas and Plemelj formula for bihypermonogenic functions presented in [15], we study a nonlinear Riemann boundary value problem for bihypermongenic functions. We first review briefly the integrals of quasi-Cauchy’s type and Plemelj formula for bihypermonogenic functions and then prove the existence of solutions of a nonlinear Riemann boundary value problem and derive the unique solution of the corresponding linear Riemann boundary value problem.

2. Preliminaries Let 𝐶ℓ0,𝑛 be a real Clifford algebra over an 𝑛 + 1dimensional real vector space 𝑅𝑛+1 with orthogonal basis 𝑒 := {𝑒0 , 𝑒1 , . . . , 𝑒𝑛 }, satisfying the relation 𝑒𝑖 𝑒𝑗 + 𝑒𝑗 𝑒𝑖 = −2𝛿𝑖𝑗 (𝑖, 𝑗 = 1, . . . , 𝑛), where 𝛿𝑖𝑗 is the usual Kronecker delta. Then 𝐶ℓ0,𝑛 has its basis 𝑒0 = 1, 𝑒1 , . . . , 𝑒𝑛 ; 𝑒1 𝑒2 , . . . , 𝑒𝑛−1 𝑒𝑛 ; . . . ; 𝑒1 ⋅ ⋅ ⋅ 𝑒𝑛 . Hence the real Clifford algebra is formed by the elements presented as 𝑎 = ∑𝐴 𝑥𝐴 𝑒𝐴 , 𝑥𝐴 ∈ R, where 𝐴 = {𝑖1 , 𝑖2 , . . . , 𝑖𝑘 | 1 ≤ 𝑖1 < 𝑖2 < ⋅ ⋅ ⋅ < 𝑖𝑘 ≤ 𝑛} or 𝐴 = 0 and 𝑒0 = 𝑒0 . For 𝑎 ∈ 𝐶𝑙0,𝑛 , we give some calculations as follows: 󸀠 , 𝑎󸀠 = ∑𝑥𝐴 𝑒𝐴 𝐴

(1)

2

Abstract and Applied Analysis

󸀠 where 𝑒𝐴 = (−1)|𝐴| 𝑒𝐴 and |𝐴| = 𝑛𝐴 is the cardinality of 𝐴; that is, when 𝐴 = 0, |𝐴| = 0 and when 𝐴 = {𝛼1 , 𝛼2 , . . . , 𝛼ℎ } ≠ 0, then |𝐴| = ℎ; 𝑒0󸀠 = 1, 𝑒𝑖󸀠 = −𝑒𝑖 , 𝑖 = 1, . . . , 𝑛. Recall that any element 𝑥 ∈ 𝐶ℓ0,𝑛 may be uniquely decomposed as 𝑥 = 𝑏 + 𝑐𝑒𝑛 , for 𝑏, 𝑐 ∈ 𝐶ℓ0,𝑛−1 (the Clifford algebra generated by 𝑒0 , . . . , 𝑒𝑛−1 ). Using this decomposition, we define the mappings 𝑃 : 𝐶ℓ0,𝑛 → 𝐶ℓ0,𝑛−1 and 𝑄 : 𝐶ℓ0,𝑛 → 𝐶ℓ0,𝑛−1 by 𝑃𝑥 = 𝑏 and 𝑄𝑥 = 𝑐. Note that if 𝑥 = ∑𝐴 𝑥𝐴𝑒𝐴 ∈ 𝐶ℓ0,𝑛 , then

𝑃𝑥 = ∑ 𝑥𝐴𝑒𝐴 , 𝑛∉𝐴

󸀠

𝑃𝑥=

𝑛∈𝐴

󸀠

𝑄𝑥=

󸀠 . ∑ 𝑥𝐴 𝑒𝐴\{𝑛} 𝑛∈𝐴

(2)

𝑛

𝑖=0

𝑛

𝜕𝑓 , 𝜕𝑥𝑖

𝜕𝑓 𝑒𝑖 , 𝜕𝑥 𝑖 𝑖=0

𝐷𝑟 𝑓 = ∑

(3)

and the modified Dirac operator 𝑀𝑙 𝑓 (𝑥) = 𝐷𝑙 𝑓 (𝑥) + (𝑛 − 1)

𝑄󸀠 𝑓 , 𝑥𝑛

𝑄𝑓 . 𝑀 𝑓 (𝑥) = 𝐷𝑟 𝑓 (𝑥) + (𝑛 − 1) 𝑥𝑛

∑

𝐴⊂{1,...,𝑚}𝐵⊂{𝑚+1,...,𝑚+𝑘}

𝑓𝐴,𝐵 (𝑥, 𝑦) 𝑒𝐴 𝑒𝐵 ,

(5)

with values in 𝐶ℓ0,𝑘+𝑚 for which 𝑓𝐴,𝐵 (𝑥, 𝑦) ∈ C𝑟 (Ω). 𝑚+1 Definition 1. Let 𝑓 ∈ F(1) \ {𝑥𝑚 = 0}, 𝑦 ∈ Ω and 𝑥 ∈ R 𝑘+1 R \{𝑦𝑘 = 0}. A function 𝑓(𝑥, 𝑦) is called bihypermonogenic on Ω, if

𝑀𝑥𝑙 𝑓 (𝑥, 𝑦) = 0,

𝑀𝑦𝑟 𝑓 (𝑥, 𝑦) = 0,

(6)

for any (𝑥, 𝑦) ∈ Ω, where 𝑀𝑥𝑙 𝑓 (𝑥, 𝑦) =

𝑚 𝜕𝑓 𝜕𝑓 (𝑥, 𝑦) + ∑𝑒𝑖 (𝑥, 𝑦) 𝜕𝑥0 𝑖=1 𝜕𝑥𝑖

𝑄󸀠 (𝑓 (𝑥, 𝑦)) + (𝑚 − 1) 𝑥 𝑥𝑚

𝑘 𝜕𝑓 𝜕𝑓 (𝑥, 𝑦) + ∑ (𝑥, 𝑦) 𝑒𝑖+𝑚 𝜕𝑦0 𝑖=1 𝜕𝑦𝑖

+ (𝑘 − 1)

𝑄𝑦 (𝑓 (𝑥, 𝑦)) 𝑦𝑘

𝐴 𝑚+𝑘∈𝐵

In this section, we give some simple review on the Cauchy integral formula and Plemelj formula for bihypermonogenic functions obtained by us and presented in [15]. We first give some notations which will be used in the following analysis. A function 𝑓(𝑥, 𝑦) : 𝜕Ω1 × 𝜕Ω2 → 𝐶𝑙0,𝑛 is said to be Hölder continuous on 𝜕Ω1 × 𝜕Ω2 , if 𝑓(𝑥, 𝑦) satisfies

(𝑥1 , 𝑦1 ) , (𝑥2 , 𝑦2 ) ∈ 𝜕Ω1 × 𝜕Ω2 ,

(0 < 𝛽 < 1) .

(10)

Denote by 𝐻(𝜕Ω1 × 𝜕Ω2 , 𝛽) the set of all Hölder continuous functions on 𝜕Ω1 × 𝜕Ω2 with the index 𝛽. For any 𝑓 ∈ 𝐻(𝜕Ω1 × 𝜕Ω2 , 𝛽), define the norm in 𝐻(𝜕Ω1 × 𝜕Ω2 , 𝛽) as ‖𝑓‖𝛽 = 𝐶(𝑓, 𝜕Ω1 × 𝜕Ω2 ) + 𝐻(𝑓, 𝜕Ω1 × 𝜕Ω2 , 𝛽), where

𝐶 (𝑓, 𝜕Ω1 × 𝜕Ω2 ) =

󵄨󵄨 󵄨 󵄨󵄨𝑓 (𝑢1 , V1 ) − 𝑓 (𝑢2 , V2 )󵄨󵄨󵄨 󵄨 󵄨𝛽 , (𝑢𝑖 ,𝜐𝑖 )∈𝜕Ω1 ×𝜕Ω2 󵄨󵄨(𝑢1 , V1 ) − (𝑢2 , V2 )󵄨󵄨 󵄨 󵄨 sup

(𝑢1 ,V1 ) ≠ (𝑢2 ,𝜐2 )

𝐻 (𝑓, 𝜕Ω1 × 𝜕Ω2 , 𝛽) =

max

(𝑢,𝜐)∈𝜕Ω1 ×𝜕Ω2

󵄨 󵄨󵄨 󵄨󵄨𝑓 (𝑢, 𝜐)󵄨󵄨󵄨 ,

𝑓 ∈ 𝐻 (𝜕Ω1 × 𝜕Ω2 , 𝛽) . (11) (7) Furthermore, for any 𝑓, 𝑔 ∈ 𝐻(𝜕Ω1 × 𝜕Ω2 , 𝛽), we have

is the left modified Dirac operator in 𝐶ℓ0,𝑚 calculated with respect to 𝑥 ∈ R𝑚+1 \ {𝑥𝑚 = 0} and 𝑀𝑦𝑟 𝑓 (𝑥, 𝑦) =

(9)

𝑄𝑦 𝑓 (𝑥, 𝑦) = ∑ ∑ 𝑓𝐴,𝐵 (𝑥, 𝑦) 𝑒𝐴 𝑒𝐵\{𝑚+𝑘} .

󵄨󵄨 󵄨 󵄨 󵄨𝛽 󵄨󵄨𝑓 (𝑥1 , 𝑦1 ) − 𝑓 (𝑥2 , 𝑦2 )󵄨󵄨󵄨 ≤ 𝑀1 󵄨󵄨󵄨(𝑥1 , 𝑦1 ) − (𝑥2 , 𝑦2 )󵄨󵄨󵄨 ,

Denote by Ω = Ω1 × Ω2 an open connected set in the Euclidean space R𝑚+1 × R𝑘+1 , 1 ≤ 𝑚 ≤ 𝑛, 1 ≤ 𝑘 ≤ 𝑛. Define a set F(𝑟) Ω to consist of all functions ∑

= ∑ ∑(−1)|𝐴\{𝑚}| 𝑓𝐴,𝐵 (𝑥, 𝑦) 𝑒𝐴\{𝑚} 𝑒𝐵 ,

(4)

𝑟

𝑓 (𝑥, 𝑦) =

𝑚∈𝐴 𝐵

3. The Cauchy Integral Formula and Plemelj Formula

We also introduce the Dirac operator 𝐷𝑙 𝑓 = ∑𝑒𝑖

󸀠 𝑄𝑥󸀠 𝑓 (𝑥, 𝑦) = ∑ ∑𝑓𝐴,𝐵 (𝑥, 𝑦) 𝑒𝐴\{𝑚} 𝑒𝐵

𝑚∈𝐴 𝐵

𝑄𝑥 = ∑ 𝑥𝐴𝑒𝐴\{𝑛} ,

󸀠 , ∑ 𝑥𝐴 𝑒𝐴 𝑛∉𝐴

is the right modified Dirac operator in the Clifford algebra generated by 𝑒0 , 𝑒𝑚+1 , . . . , 𝑒𝑚+𝑘 calculated with respect to 𝑦 ∈ R𝑘+1 \ {𝑦𝑘 = 0}, where

(8)

󵄩 󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩𝑓𝑔󵄩󵄩𝛽 ≤ 𝐽0 󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝛽 󵄩󵄩󵄩𝑔󵄩󵄩󵄩𝛽 .

(12)

Theorem 2 (see [15]). Let Ω󸀠 and Ω󸀠󸀠 be open subsets of R𝑚+1 + 󸀠 and R𝑘+1 + , respectively. Suppose that Ω1 and Ω2 satisfy Ω1 ⊂ Ω and Ω2 ⊂ Ω󸀠󸀠 , respectively. The boundaries 𝜕Ω1 , 𝜕Ω2 of Ω1 , Ω2 are differentiable, oriented, compact Liapunov surfaces. If


3

𝜑(𝑥, 𝑦) is a bihypermonogenic function in Ω󸀠 ×Ω󸀠󸀠 , 𝑥 ∈ Ω1 , 𝑦 ∈ Ω2 , then

Definition 3 (see [15]). The integral 𝜙 (𝑡1 , 𝑡2 )

𝜑 (𝑥, 𝑦) = 𝜆∫

𝜕Ω1 ×𝜕Ω2

𝐸𝑚 (𝑢, 𝑥) 𝑑𝜎𝑚 (𝑢) 𝜑 (𝑢, V) 𝑑𝜎𝑘 (V) E𝑘 (V, 𝑦)

= 𝜆∫

𝜕Ω1 ×𝜕Ω2

𝐸𝑚 (𝑢, 𝑡1 ) 𝑑𝜎𝑚 (𝑢) 𝜑 (𝑢, V) 𝑑𝜎𝑘 (V) E𝑘 (V, 𝑡2 )

− 𝜆∫

̃ ̃ V)𝑑𝜎 𝐸𝑚 (𝑢, 𝑥) 𝑑𝜎𝑚 (𝑢) 𝜑(𝑢, 𝑘 (V)F𝑘 (V, 𝑦)

− 𝜆∫

̃ ̃ V)d𝜎 𝐸𝑚 (𝑢, 𝑡1 ) 𝑑𝜎𝑚 (𝑢) 𝜑(𝑢, 𝑘 (V)F𝑘 (V, 𝑡2 )

− 𝜆∫

̂ ̂ 𝐹𝑚 (𝑢, 𝑥) 𝑑𝜎 𝑚 (𝑢)𝜑(𝑢, V)𝑑𝜎𝑘 (V) E𝑘 (V, 𝑦)

− 𝜆∫

̂ ̂ 𝐹𝑚 (𝑢, 𝑡1 ) 𝑑𝜎 𝑚 (𝑢)𝜑(𝑢, V)𝑑𝜎𝑘 (V) E𝑘 (V, 𝑡2 )

+ 𝜆∫

̃ ̂ ̂̃ 𝐹𝑚 (𝑢, 𝑥) 𝑑𝜎 𝑚 (𝑢)𝜑(𝑢, V)𝑑𝜎𝑘 (V)F𝑘 (V, 𝑦) ,

+ 𝜆∫

̃ ̂ ̂̃ 𝐹𝑚 (𝑢, 𝑡1 ) 𝑑𝜎 𝑚 (𝑢)𝜑(𝑢, V)𝑑𝜎𝑘 (V)F𝑘 (V, 𝑡2 )

𝜕Ω1 ×𝜕Ω2

𝜕Ω1 ×𝜕Ω2

𝜕Ω1 ×𝜕Ω2

𝜕Ω1 ×𝜕Ω2

𝜕Ω1 ×𝜕Ω2

(16)

(13)

where

is called a singular integral on 𝜕Ω1 × 𝜕Ω2 , where 𝜆, 𝐸𝑚 (𝑢, 𝑡1 ), E𝑘 (V, 𝑡2 ), 𝐹𝑚 (𝑢, 𝑡1 ), and F𝑘 (V, 𝑡1 ) are given in Theorem 2.

𝑑𝜎𝑚 = 𝑑𝑥1 ∧ 𝑑𝑥2 ∧ ⋅ ⋅ ⋅ 𝑑𝑥𝑚 𝑚

+ ∑(−1)𝑖 𝑒𝑖 𝑑𝑥0 ∧ ⋅ ⋅ ⋅ ∧ 𝑑𝑥𝑖−1 ∧ 𝑑𝑥𝑖+1 ∧ ⋅ ⋅ ⋅ 𝑑𝑥𝑚 , 𝑖=1

𝑑𝜎𝑘 = 𝑑𝑦1 ∧ 𝑑𝑦2 ∧ ⋅ ⋅ ⋅ 𝑑𝑦𝑘 𝑘

𝜕Ω1 ×𝜕Ω2

Definition 4 (see [15]). Let 𝛿 > 0 be a constant and 𝜆 𝛿 = 𝐵1 (𝑡1 , 𝛿) × 𝐵2 (𝑡2 , 𝛿), where 𝐵𝑖 (𝑡𝑖 , 𝛿) (𝑖 = 1, 2) are balls with the center at 𝑡𝑖 and the radius 𝛿 > 0. Denote 𝜙𝛿 (𝑡1 , 𝑡2 )

𝑖

+ ∑(−1) 𝑒𝑖+𝑚 𝑑𝑦0 ∧ 𝑑𝑦1 ∧ ⋅ ⋅ ⋅ ∧ 𝑑𝑦𝑖−1 ∧ 𝑑𝑦𝑖+1

= 𝜆∫

𝑖=1

𝐸𝑚 (𝑢, 𝑡1 ) 𝑑𝜎𝑚 (𝑢) 𝜑 (𝑢, V) 𝑑𝜎𝑘 (V) E𝑘 (V, 𝑡2 )

𝜕Ω1 ×𝜕Ω2 \𝜆 𝛿

∧ ⋅ ⋅ ⋅ 𝑑𝑦𝑘 ,

𝑦 = 𝑦0 + 𝑦1 𝑒𝑚+1 + ⋅ ⋅ ⋅ + 𝑦𝑘 𝑒𝑚+𝑘 ,

𝐸𝑙 (𝑢, 𝑥) = 𝐹𝑙 (𝑢, 𝑥) =

(𝑢 − 𝑥)

̂ 𝑙−1 |𝑢 − 𝑥|𝑙−1 |𝑢 − 𝑥|

If lim𝛿 → 0 𝜙𝛿 (𝑡1 , 𝑡2 ) = 𝐼, then 𝐼 is called the Cauchy principal value of a singular integral, denoted by 𝐼 = 𝜙(𝑡1 , 𝑡2 ).

𝑢, 𝑥 ∈ R𝑚+1 ,

,

= {𝑥 = Lemma 5 (see [11]). Let Ω be an open subset of R𝑛+1 + (𝑥0 , 𝑥1 , . . . , 𝑥𝑛 ) | 𝑥𝑛 > 0} and let 𝐾 be an 𝑛 + 1-chain satisfying 𝐾 ⊂ Ω; then 2𝑛−1 𝑦𝑛𝑛−1 ̂ (∫ 𝐸𝑛 (𝑥, 𝑦) 𝑑𝜎 (𝑥) − ∫ 𝐹𝑛 (𝑥, 𝑦) 𝑑𝜎(𝑥)) 𝜔𝑛+1 𝜕𝐾 𝜕𝐾

−1

(̃V − 𝑦) , F𝑙 (V, 𝑦) = 󵄨 󵄨󵄨V − 𝑦󵄨󵄨󵄨𝑙−1 󵄨󵄨󵄨V − 𝑦̃󵄨󵄨󵄨𝑙−1 󵄨 󵄨 󵄨 󵄨

(14)

and the involutions ̂ and ̃ are defined by

𝑒̃𝑖 = 𝑒𝑖 ,

𝑖 ∈ {0, 1, . . . , 𝑚 + 𝑘} \ {𝑚} , ̂ = 𝑎̂̂𝑏, 𝑎𝑏

𝑖 ∈ {0, 1, . . . , 𝑚 + 𝑘 − 1} ,

𝑒̃ 𝑚+𝑘 = −𝑒𝑚+𝑘 ,

̃ = 𝑎̃̃𝑏. 𝑎𝑏

̃ ̂ ̂̃ 𝐹𝑚 (𝑢, 𝑡1 ) 𝑑𝜎 𝑚 (𝑢)𝜑(𝑢, V)d𝜎𝑘 (V)F𝑘 (V, 𝑡2 ) . (17)

−1

𝑒̂ 𝑚 = −𝑒𝑚 ,

+ 𝜆∫

𝜕Ω1 ×𝜕Ω2 \𝜆 𝛿

(V − 𝑦) , E𝑙 (V, 𝑦) = 󵄨 󵄨󵄨V − 𝑦󵄨󵄨󵄨𝑙−1 󵄨󵄨󵄨V − 𝑦̃󵄨󵄨󵄨𝑙−1 󵄨 󵄨 󵄨 󵄨

𝑒̂𝑖 = 𝑒𝑖 ,

̂ ̂ 𝐹𝑚 (𝑢, 𝑡1 ) 𝑑𝜎 𝑚 (𝜇)𝜑(𝑢, V)𝑑𝜎𝑘 (V) E𝑘 (V, 𝑡2 )

𝑢, 𝑥 ∈ R𝑚+1 ,

, ̂ 𝑙−1 |𝑢 − 𝑥|𝑙−1 |𝑢 − 𝑥| 𝑢 − 𝑥)−1 (̂

− 𝜆∫

𝜕Ω1 ×𝜕Ω2 \𝜆 𝛿

𝑚−1 𝑘−1 𝑘−1 2 𝑦𝑘 2𝑚−1 𝑥𝑚 , 𝜔𝑚+1 𝜔𝑘+1 −1

̃ ̃ V)d𝜎 𝐸𝑚 (𝑢, 𝑡1 ) 𝑑𝜎𝑚 (𝑢) 𝜑(𝑢, 𝑘 (V)F𝑘 (V, 𝑡2 )

𝜕Ω1 ×𝜕Ω2 \𝜆 𝛿

V = V0 + V1 𝑒𝑚+1 + ⋅ ⋅ ⋅ + V𝑘 𝑒𝑚+𝑘 ,

𝜆=

− 𝜆∫

1, ={ 0,

𝑦 ∈ 𝐾, 𝑦 ∈ R𝑛+1 + − 𝐾.

(18)

Lemma 6 (see [11]). Let Ω, 𝐾 and 𝜕𝐾 be as in Lemma 5 and 𝑦 ∈ 𝜕𝐾; then (15)

2𝑛−1 𝑦𝑛𝑛−1 ̂ = 1. (∫ 𝐸𝑛 (𝑥, 𝑦) 𝑑𝜎 (𝑥) − ∫ 𝐹𝑛 (𝑥, 𝑦) 𝑑𝜎(𝑥)) 𝜔𝑛+1 2 𝜕𝐾 𝜕𝐾 (19)

4


Theorem 7 (see [15]). If 𝜑(𝑢, V) ∈ 𝐻(𝜕Ω1 ×𝜕Ω2 , 𝛽), then there exists the Cauchy principal value of singular integrals and 𝜙 (𝑡1 , 𝑡2 ) 1 = − 𝜑 (𝑡1 , 𝑡2 ) + 𝑋1 (𝑡1 , 𝑡2 ) + 𝑋2 (𝑡1 , 𝑡2 ) + 𝑋3 (𝑡1 , 𝑡2 ) 4 1 1 + 𝑋4 (𝑡1 , 𝑡2 ) + (𝑃1 𝜑 + 𝑃2 𝜑) + (𝑄1 𝜑 + 𝑄2 𝜑) , 4 4

(20)

\ Ω1 , Ω−2 = R𝑘+1 Set Ω+𝑖 = Ω𝑖 , (𝑖 = 1, 2), Ω−1 = R𝑚+1 + + \ Ω2 , ± and denote 𝑥(∈ Ω1 ) → 𝑡1 ∈ 𝜕Ω1 by 𝑥 → 𝑡1± . Moreover denote 𝑦 (∈ Ω±2 ) → 𝑡2 ∈ 𝜕Ω2 by 𝑦 → 𝑡2± and denote by 𝜙±± (𝑡1 , 𝑡2 ) the limits of 𝜙(𝑥, 𝑦) when (𝑥, 𝑦) → (𝑡1± , 𝑡2± ). Then we have the following important theorem. Theorem 8 (see [15]). If 𝜑(𝑢, V) ∈ 𝐻(𝜕Ω1 × 𝜕Ω2 , 𝛽), then 𝜙++ (𝑡1 , 𝑡2 ) =

where 𝑋1 (𝑡1 , 𝑡2 )

1 [𝜑 (𝑡1 , 𝑡2 ) + 𝑃1 (𝜑) + 𝑃2 (𝜑) + 𝑄1 (𝜑) + 𝑄2 (𝜑) 4 +𝑃3 (𝜑)] ,

= 𝜆∫

𝜙+− (𝑡1 , 𝑡2 )

𝐸𝑚 (𝑢, 𝑡1 ) 𝑑𝜎𝑚 (𝑢) 𝜓1 (𝑢, V) 𝑑𝜎𝑘 (V)

𝜕Ω1 ×𝜕Ω2

=

× E𝑘 (V, 𝑡2 ) , 𝑋2 (𝑡1 , 𝑡2 )

1 [−𝜑 (𝑡1 , 𝑡2 ) − 𝑃1 (𝜑) + 𝑃2 (𝜑) − 𝑄1 (𝜑) + 𝑄2 (𝜑) 4 +𝑃3 (𝜑)] ,

𝜙−+ (𝑡1 , 𝑡2 )

̃ 𝐸𝑚 (𝑢, 𝑡1 ) 𝑑𝜎𝑚 (𝑢) 𝜓2 (𝑢, V) 𝑑𝜎 𝑘 (V)

= −𝜆 ∫

𝜕Ω1 ×𝜕Ω2

=

× F𝑘 (V, 𝑡2 ) , 𝑋3 (𝑡1 , 𝑡2 )

1 [−𝜑 (𝑡1 , 𝑡2 ) + 𝑃1 (𝜑) − 𝑃2 (𝜑) + 𝑄1 (𝜑) − 𝑄2 (𝜑) 4 +𝑃3 (𝜑)] ,

𝜙−− (𝑡1 , 𝑡2 )

̂ 𝐹𝑚 (𝑢, 𝑡1 ) 𝑑𝜎 𝑚 (𝑢)𝜓3 (𝑢, V) 𝑑𝜎𝑘 (V)

= −𝜆 ∫

𝜕Ω1 ×𝜕Ω2

=

× E𝑘 (V, 𝑡2 ) ,

1 [𝜑 (𝑡1 , 𝑡2 ) − 𝑃1 (𝜑) − 𝑃2 (𝜑) − 𝑄1 (𝜑) − 𝑄2 (𝜑) 4 +𝑃3 (𝜑)] ,

𝑋4 (𝑡1 , 𝑡2 ) = 𝜆∫

𝜕Ω1 ×𝜕Ω2

(22)

̃ ̂ 𝐹𝑚 (𝑢, 𝑡1 ) 𝑑𝜎 𝑚 (𝑢)𝜓4 (𝑢, V) 𝑑𝜎𝑘 (V) (21)

× F𝑘 (V, 𝑡2 ) ,

where (𝑡1 , 𝑡2 ) ∈ 𝜕Ω1 × 𝜕Ω2 , 𝑃3 (𝜑) = 4 𝜙 (𝑡1 , 𝑡2 ).

𝜓1 (𝑢, V) = 𝜑 (𝑢, V) − 𝜑 (𝑢, 𝑡2 ) − 𝜑 (𝑡1 , V) + 𝜑 (𝑡1 , 𝑡2 ) ,

4. The Boundary Value Problem for Bihypermonogenic Functions

(𝑡1 , V) + 𝜑 (𝑡1 , 𝑡2 ) , 𝜓2 (𝑢, V) = 𝜑̃ (𝑢, V) − 𝜑 (𝑢, 𝑡2 ) − 𝜑̃

In this section, we consider the boundary value problem.

𝜓3 (𝑢, V) = 𝜑̂ (𝑢, 𝑡2 ) − 𝜑 (𝑡1 , V) + 𝜑 (𝑡1 , 𝑡2 ) , (𝑢, V) − 𝜑̂ ̃ (𝑢, 𝑡2 ) − 𝜑̃ (𝑡1 , V) + 𝜑 (𝑡1 , 𝑡2 ) , 𝜓4 (𝑢, V) = 𝜑̂ (𝑢, V) − 𝜑̂ 𝑃1 𝜑 = 2𝜆 1 ∫

𝜕Ω1

𝑃2 𝜑 = 2𝜆 2 ∫

𝐸𝑚 (𝑢, 𝑡1 ) 𝑑𝜎𝑚 (𝑢) 𝜑 (𝑢, 𝑡2 ) ,

𝜕Ω2

𝜑 (𝑡1 , V) 𝑑𝜎𝑘 (V) E𝑘 (V, 𝑡2 ) ,

𝑄1 𝜑 = −2𝜆 1 ∫

̂ ̂ 𝐹𝑚 (𝑢, 𝑡1 ) 𝑑𝜎 𝑚 (𝑢)𝜑 (𝑢, 𝑡2 ),

𝑄2 𝜑 = −2𝜆 2 ∫

̃ 𝜑̃ (𝑡1 , 𝜐) 𝑑𝜎 𝑘 (V)F𝑘 (V, 𝑡2 ) ,

𝜕Ω1

𝜕Ω2

𝜆 = 𝜆1𝜆2,

𝜆1 =

𝑚−1 2𝑚−1 𝑥𝑚 , 𝜔𝑚+1

𝜆2 =

2𝑘−1 𝑦𝑘𝑘−1 . 𝜔𝑘+1

Definition 9. Let Ω1 × Ω2 and 𝐻(𝜕Ω1 × 𝜕Ω2 , 𝛽) be as before. We want to find a bihypermonogenic function 𝜙(𝑥, 𝑦) defined in R𝑚+1 × R𝑘+1 + + /𝜕Ω1 × 𝜕Ω2 , which is continuous to 𝜕Ω1 × 𝜕Ω2 +− and 𝜙 (𝑥, ∞) = 𝜙−+ (∞, 𝑦) = 𝜙−− (∞, ∞) = 0 and satisfies the nonlinear boundary condition 𝐴 (𝑡1 , 𝑡2 ) 𝜙++ (𝑡1 , 𝑡2 ) + 𝐵 (𝑡1 , 𝑡2 ) 𝜙+− (𝑡1 , 𝑡2 ) + 𝐶 (𝑡1 , 𝑡2 ) 𝜙−+ (𝑡1 , 𝑡2 ) + 𝐷 (𝑡1 , 𝑡2 ) 𝜙−− (𝑡1 , 𝑡2 ) = 𝑔 (𝑡1 , 𝑡2 ) 𝑓 (𝑡1 , 𝑡2 , 𝜙++ (𝑡1 , 𝑡2 ) , 𝜙+− (𝑡1 , 𝑡2 ) , 𝜙−+ (𝑡1 , 𝑡2 ) , 𝜙−− (𝑡1 , 𝑡2 )) , (23) in which 𝐴(𝑡1 , 𝑡2 ), 𝐵(𝑡1 , 𝑡2 ), 𝐶(𝑡1 , 𝑡2 ), 𝐷(𝑡1 , 𝑡2 ), 𝑔(𝑡1 , 𝑡2 ) ∈ 𝐻(𝜕Ω1 × 𝜕Ω2 , 𝛽) and 𝑓 are known functions. The above boundary value problem is called Problem R.


5

From Theorem 8, we can transform the boundary condition of Problem R into an integral equation 𝐹𝜑 = 𝜑,

𝑃2 𝜑 + 𝑄2 𝜑 = 2𝜆 2 ∫

𝜕Ω2

(24)

𝜑 (𝑡1 , 𝜐) 𝑑𝜎𝑘 (𝜐) 𝐸𝑘 (𝜐, 𝑡2 )

− 2𝜆 2 ∫

𝜕Ω2

where

̃ ̃ 𝜑(𝑡 1 , 𝜐)𝑑𝜎𝑘 (𝜐)𝐹𝑘 (𝜐, 𝑡2 ) (30)

𝐹𝜑

and Theorem 10, we can obtain the result. = (𝐴 + 𝐵) (𝜑 + 𝑃1 𝜑 + 𝑃2 𝜑 + 𝑄1 𝜑 + 𝑄2 𝜑 + 𝑃3 𝜑) + (𝐶 + 𝐷) (−𝜑 + 𝑃1 𝜑 − 𝑃2 𝜑 + 𝑄1 𝜑 − 𝑄2 𝜑 + 𝑃3 𝜑) + (𝐵 + 𝐷) (2𝜑 − 2𝑃1 𝜑 − 2𝑄1 𝜑) + (1 − 4𝐵) 𝜑 − 4𝑔𝑓. (25)

Theorem 10 (see [9]). Let Ω, 𝜕Ω ⊂ 𝑅+𝑙+1 , and let 𝐻(𝜕Ω, 𝛽) be the set of Hölder continuous functions on 𝜕Ω with the index 𝛽. For 𝜑 ∈ 𝐻(𝜕Ω, 𝛽) and 𝜃𝜑 = 𝜙𝜑 − 𝜙𝜑 =

𝜑 , 2

Theorem 13. Suppose the boundaries 𝜕Ω1 , 𝜕Ω2 of Ω1 , Ω2 be differentiable, oriented, compact Liapunov surfaces. If 𝜑(𝑡1 , 𝑡2 ) ∈ 𝐻(𝜕Ω1 × 𝜕Ω2 , 𝛽), then 󵄩󵄩󵄩(𝑃2 𝜑 + 𝑄2 𝜑) ± 𝑃3 𝜑󵄩󵄩󵄩 ≤ 𝐽3 󵄩󵄩󵄩𝜑󵄩󵄩󵄩 , (31) 󵄩 󵄩𝛽 󵄩 󵄩𝛽 where 𝐽3 is a positive constant which is independent of 𝜑. Proof. From (20), it follows that 4

𝑃2 𝜑 + 𝑄2 𝜑 − 𝑃3 𝜑 = 𝜑 − 4∑𝑋𝑖 (𝑡1 , 𝑡2 ) − (𝑃1 𝜑 + 𝑄1 𝜑) . (32) 𝑖=1

2𝑙−1 𝑥𝑙𝑙−1 [∫ 𝐸𝑙 (𝑡, 𝑥) 𝑑𝜎0 (𝑡) 𝜑 (𝑡) 𝜔𝑙+1 𝜕Ω

(26)

̂ ̂ − 𝐹𝑙 (𝑡, 𝑥) 𝑑𝜎 0 (𝑡)𝜑 (𝑡)] , where 𝐸𝑙 (𝑡, 𝑥), 𝐹𝑙 (𝑡, 𝑥) are as before, then 𝜃𝜑 is a hypermonogenic function with 󵄩 󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩𝜃𝜑󵄩󵄩𝛽 ≤ 𝐽1 󵄩󵄩󵄩𝜑󵄩󵄩󵄩𝛽 , (27) where 𝐽1 is a constant independent of 𝜑. Lemma 11 (see [6]). Let 𝐸𝑚 (𝑢, 𝑥), 𝐸𝑚 (𝑢, 𝑡1 ) be the same as in Theorem 2. If 𝑢 ∈ Ω1 , 𝑥 ∈ Ω2 , 𝑡1 ∈ Ω2 , then there exists a constant 𝑀 such that 󵄨 󵄨󵄨 󵄨󵄨𝐸𝑚 (𝑢, 𝑥) − 𝐸𝑚 (𝑢, 𝑡1 )󵄨󵄨󵄨

Moreover, based on Lemma 12 we only need to prove ‖ ∑4𝑖=1 𝑋𝑖 (𝑡1 , 𝑡2 )‖𝛽 ≤ 𝐽4 ‖𝜑‖𝛽 . It is easy to

prove | ∑4𝑖=1 𝑋𝑖 (𝑡1 , 𝑡2 )| ≤ 𝐵1 ‖𝜑‖𝛽 . We rewrite 𝜓𝑖 (𝑢, V) = 1, 2, 3, 4). Now we consider as 𝜓𝑖0 (𝑡1 , 𝑡2 ), (𝑖 4 𝐻(∑𝑖=1 𝑋𝑖 (𝑡1 , 𝑡2 ), 𝜕Ω1 × 𝜕Ω2 , 𝛽) and write 𝛿 = |(𝑡1 , 𝑡2 ) − (𝑡1󸀠 , 𝑡2󸀠 )| = √𝛿12 + 𝛿22 for any (𝑡1 , 𝑡2 ), (𝑡1󸀠 , 𝑡2󸀠 ) ∈ 𝜕Ω1 × 𝜕Ω2

󸀠 󸀠 and denote by 𝜌01 , 𝜌02 , 𝜌01 , 𝜌02 the projections of |𝜇 − 𝑡1 |, |𝜐 − 𝑡2 |, |𝜇 − 𝑡1󸀠 |, |𝜐 − 𝑡2󸀠 | on the tangent plane of 𝑡1 , 𝑡2 , 𝑡󸀠1 , 𝑡2󸀠 , respectively. Moreover we construct spheres 𝑂𝑖 (𝑡𝑖 , 3𝛿𝑖 ) with the center at 𝑡𝑖 and radius 3𝛿𝑖 , where 6𝛿𝑖 < 𝑑𝑖 , 𝛿𝑖 < 1, 𝑖 = 1, 2, where 𝑑𝑖 is a constant as in [5]. Denote by 𝜕Ω𝑖1 , 𝜕Ω𝑖2 the part of 𝜕Ω𝑖 lying inside the sphere 𝑂𝑖 and its surplus part, respectively, and set 4

4

𝑖=1

𝑖=1

𝑅 (𝜕Ω1 × 𝜕Ω2 ) = ∑𝑋𝑖 (𝑡1 , 𝑡2 ) − ∑𝑋𝑖 (𝑡1󸀠 , 𝑡2󸀠 )

(28) 󵄨 𝑢 − 𝑡1 󵄨󵄨󵄨𝑖 󵄨󵄨󵄨 𝑥 − 𝑡1 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 + 󵄨󵄨𝑥 − 𝑡1 󵄨󵄨󵄨󵄨] 󵄨󵄨󵄨󵄨𝑢 − 𝑡1 󵄨󵄨󵄨󵄨−𝑚 . ≤ 𝑀 [ ∑ 󵄨󵄨󵄨󵄨 󵄨 𝑢 − 𝑥 󵄨󵄨 󵄨󵄨 𝑢 − 𝑥 󵄨󵄨 𝑚−1󵄨

4

4

𝑖=1

𝑖=1

= ∑𝑋𝑖 (𝜕Ω1 × 𝜕Ω2 ) − ∑𝑋𝑖 (𝜕Ω1 × 𝜕Ω2 ) .

𝑖=0

(33) From

Lemma 12. If 𝜑(𝑡1 , 𝑡2 ) ∈ 𝐻(𝜕Ω1 × 𝜕Ω2 , 𝛽), then 󵄩 󵄩 󵄩󵄩 󵄩 󵄩󵄩𝜑 ± (𝑃𝑖 𝜑 + 𝑄𝑖 𝜑)󵄩󵄩󵄩𝛽 ≤ 𝐽2 󵄩󵄩󵄩𝜑󵄩󵄩󵄩𝛽 , 󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩𝑃𝑖 𝜑 + 𝑄𝑖 𝜑󵄩󵄩󵄩𝛽 ≤ 𝐽2 󵄩󵄩󵄩𝜑󵄩󵄩󵄩𝛽 ,

(29) 𝑖 = 1, 2,

where 𝐽2 is a positive constant. Proof. Using the following equations: 𝑃1 𝜑 + 𝑄1 𝜑 = 2𝜆 1 ∫

𝜕Ω1

− 2𝜆 1 ∫

𝐸𝑚 (𝑢, 𝑡1 ) 𝑑𝜎𝑚 (𝑢) 𝜑 (𝜇, 𝑡2 )

𝜕Ω1

̂ ̂ 𝐹𝑚 (𝑢, 𝑡1 ) 𝑑𝜎 𝑚 (𝑢)𝜑(𝑢, 𝑡2 ),

󵄨󵄨 0 󵄨 󵄨󵄨𝜓𝑖 (𝑡1 , 𝑡2 )󵄨󵄨󵄨 ≤ 𝑀󵄩󵄩󵄩󵄩𝜑󵄩󵄩󵄩󵄩𝛽 󵄨󵄨󵄨󵄨𝑢 − 𝑡1 󵄨󵄨󵄨󵄨𝛽 , 󵄨 󵄨 󵄨󵄨 0 󵄨 󵄨󵄨𝜓𝑖 (𝑡1 , 𝑡2 )󵄨󵄨󵄨 ≤ 𝑀󵄩󵄩󵄩󵄩𝜑󵄩󵄩󵄩󵄩𝛽 󵄨󵄨󵄨󵄨𝜐 − 𝑡2 󵄨󵄨󵄨󵄨𝛽 , 󵄨 󵄨 󵄨󵄨 0 󸀠 󸀠 󵄨󵄨 󵄨 󵄨𝛽 󵄨󵄨𝜓𝑖 (𝑡1 , 𝑡2 )󵄨󵄨 ≤ 𝑀󵄩󵄩󵄩󵄩𝜑󵄩󵄩󵄩󵄩𝛽 󵄨󵄨󵄨𝑢 − 𝑡1󸀠 󵄨󵄨󵄨 , 󵄨 󵄨 󵄨 󵄨 󵄨󵄨 0 󸀠 󸀠 󵄨󵄨 󵄨 󵄨𝛽 󵄨󵄨𝜓𝑖 (𝑡1 , 𝑡2 )󵄨󵄨 ≤ 𝑀󵄩󵄩󵄩󵄩𝜑󵄩󵄩󵄩󵄩𝛽 󵄨󵄨󵄨𝜐 − 𝑡2󸀠 󵄨󵄨󵄨 , 󵄨 󵄨 󵄨 󵄨 𝑖 = 1, 2, 3, 󵄨󵄨 0 󵄨 󵄨󵄨𝜓4 (𝑡1 , 𝑡2 )󵄨󵄨󵄨 ≤ 𝑀󵄩󵄩󵄩󵄩𝜑󵄩󵄩󵄩󵄩𝛽 , 󵄨 󵄨 󵄨󵄨󵄨𝜓0 (𝑡󸀠 , 𝑡󸀠 )󵄨󵄨󵄨 ≤ 𝑀󵄩󵄩󵄩𝜑󵄩󵄩󵄩 , 󵄨󵄨 4 1 2 󵄨󵄨 󵄩 󵄩𝛽

(34)

6


we obtain that 󵄨󵄨 󵄨 󵄨󵄨𝐸𝑚 (𝑢, 𝑡1 ) 𝑑𝜎𝑚 (𝑢) 𝜓10 (𝑡1 , 𝑡2 ) 𝑑𝜎𝑘 (𝜐) 𝐸𝑘 (𝜐, 𝑡2 )󵄨󵄨󵄨 󵄨 󵄨 (𝛽/2)−1 󵄩 󵄩 (𝛽/2)−1 ≤ 𝑀󵄩󵄩󵄩𝜑󵄩󵄩󵄩𝛽 𝜌01 𝑑𝜌01 𝜌02 𝑑𝜌02 , 󵄨󵄨 󵄨󵄨 ̃ 󵄨󵄨𝐸𝑚 (𝑢, 𝑡1 ) 𝑑𝜎𝑚 (𝑢) 𝜓20 (𝑡1 , 𝑡2 ) 𝑑𝜎 𝑘 (𝜐)𝐹𝑘 (𝜐, 𝑡2 )󵄨󵄨󵄨 󵄨 󵄩 󵄩 𝛽−1 ≤ 𝑀󵄩󵄩󵄩𝜑󵄩󵄩󵄩𝛽 𝜌01 𝑑𝜌01 𝑑𝜌02 , 󵄨󵄨 󵄨󵄨 0 ̂ 󵄨󵄨𝐹𝑚 (𝑢, 𝑡1 ) 𝑑𝜎 𝑚 (𝑢)𝜓3 (𝑡1 , 𝑡2 ) 𝑑𝜎𝑘 (𝜐) 𝐸𝑘 (𝜐, 𝑡2 )󵄨󵄨󵄨 󵄨 𝛽−1 󵄩 󵄩 ≤ 𝑀󵄩󵄩󵄩𝜑󵄩󵄩󵄩𝛽 𝑑𝜌01 𝜌02 𝑑𝜌02 , 󵄨󵄨 󵄨󵄨 0 ̂ ̃ 󵄨󵄨𝐹𝑚 (𝑢, 𝑡1 ) 𝑑𝜎 𝑚 (𝑢)𝜓4 (𝑡1 , 𝑡2 ) 𝑑𝜎𝑘 (𝜐)𝐹𝑘 (𝜐, 𝑡2 )󵄨󵄨󵄨 󵄨 󵄩 󵄩 ≤ 𝑀󵄩󵄩󵄩𝜑󵄩󵄩󵄩𝛽 𝑑𝜌01 𝑑𝜌02 .

Similarly, we can deal with (35)

𝑋3 (𝜕Ω12 × 𝜕Ω22 ) − 𝑋3 (𝜕Ω12 × 𝜕Ω22 ) = 𝐺1 + 𝐺2 + 𝐺3 , (36)

(37)

(38)

Thus we have 󵄨󵄨 󵄨 󵄨󵄨𝑅 (𝜕Ω11 × 𝜕Ω21 )󵄨󵄨󵄨 󵄨 󵄨 4 󵄨󵄨 󵄨󵄨 ≤ ∑ 󵄨󵄨󵄨󵄨𝑋𝑖 (𝜕Ω11 × 𝜕Ω21 )󵄨󵄨󵄨󵄨 + ∑ 󵄨󵄨󵄨𝑋𝑖 (𝜕Ω11 × 𝜕Ω21 )󵄨󵄨󵄨 󵄨 󵄨 𝑖=1

𝑋4 (𝜕Ω12 × 𝜕Ω22 ) − 𝑋4 (𝜕Ω12 × 𝜕Ω22 ) = 𝐻1 + 𝐻2 + 𝐻3 . (43) From Lemma 11 and (35) and by |𝜐 − 𝑡2 | ≥ 3𝛿2 > 𝛽 0, |𝜐 − 𝑡2󸀠 | ≥ 2𝛿2 , we obtain |𝐸2 | ≤ 𝐶1 ‖𝜑‖𝛽 |(𝑡1 , 𝑡2 ) − (𝑡1󸀠 , 𝑡2󸀠 )| . Similarly, we can get the inequality estimation for 𝐸1 and 𝐺2 . By (36) and (37), |𝜐 − 𝑡2 | ≥ 3𝛿2 > 0, |𝜐 − 𝑡2󸀠 | ≥ 2𝛿2 , we have 𝛽

|𝐹2 | ≤ 𝐶2 ‖𝜑‖𝛽 |(𝑡1 , 𝑡2 ) − (𝑡1󸀠 , 𝑡2󸀠 )| . Similarly, we can obtain the inequality estimation for 𝐹1 , 𝐺1 , 𝐻1 , and 𝐻2 . Since 𝐸3 + 𝐹3 + 𝐺3 + 𝐻3

4

𝑖=1

𝑋2 (𝜕Ω12 × 𝜕Ω22 ) − 𝑋2 (𝜕Ω12 × 𝜕Ω22 ) = 𝐹1 + 𝐹2 + 𝐹3 ,

(39)

󵄨𝛽 󵄩 󵄩 󵄨 ≤ 𝐵2 󵄩󵄩󵄩𝜑󵄩󵄩󵄩𝛽 󵄨󵄨󵄨󵄨(𝑡1 , 𝑡2 ) − (𝑡1󸀠 , 𝑡2󸀠 )󵄨󵄨󵄨󵄨 .

1 = 𝜆 1 ∫ 𝐸𝑚 (𝑢, 𝑡1󸀠 ) 𝑑𝜎𝑚 (𝑢) [𝜑 (𝑢, 𝑡2󸀠 ) − 𝜑 (𝑢, 𝑡2 )] 2 𝜕Ω12 1 ̂ ̂󸀠 ̂ − 𝜆 1 ∫ 𝐹𝑚 (𝑢, 𝑡1󸀠 ) 𝑑𝜎 𝑚 (𝑢) [𝜑 (𝑢, 𝑡2 ) − 𝜑 (𝑢, 𝑡2 )] 2 𝜕Ω12

Noting that |𝜐 − 𝑡2󸀠 | ≥ 2𝛿2 , |𝜐 − 𝑡2 | ≥ 3𝛿2 > 0 on 𝜕Ω22 , we have 󵄨󵄨󵄨𝑅 (𝜕Ω11 × 𝜕Ω22 )󵄨󵄨󵄨 ≤ 𝐵3 󵄩󵄩󵄩𝜑󵄩󵄩󵄩 󵄨󵄨󵄨󵄨(𝑡1 , 𝑡2 ) − (𝑡󸀠 , 𝑡󸀠 )󵄨󵄨󵄨󵄨𝛽 . (40) 1 2 󵄨 󵄨 󵄩 󵄩𝛽 󵄨 󵄨 Similarly, we can obtain |𝑅(𝜕Ω12 × 𝜕Ω21 )| 𝛽 𝐵4 ‖𝜑‖𝛽 |(𝑡1 , 𝑡2 ) − (𝑡1󸀠 , 𝑡2󸀠 )| . Next we want to prove 󵄩 󵄩 󵄨 󵄨 󵄨󵄨 󸀠 󸀠 󵄨𝛽 󵄨󵄨𝑅 (𝜕Ω12 × 𝜕Ω22 )󵄨󵄨󵄨 ≤ 𝐶4 󵄩󵄩󵄩𝜑󵄩󵄩󵄩𝛽 󵄨󵄨󵄨󵄨(𝑡1 , 𝑡2 ) − (𝑡1 , 𝑡2 )󵄨󵄨󵄨󵄨 .

1 + 𝜆 2 ∫ [𝜑 (𝑡1󸀠 , 𝜐) − 𝜑 (𝑡1 , 𝜐)] 𝑑𝜎𝑘 (𝜐) 𝐸𝑘 (𝜐, 𝑡2󸀠 ) 2 𝜕Ω22

≤

1 󸀠 󸀠 ̃ ̃ ̃ − 𝜆 2 ∫ [𝜑(𝑡 1 , 𝜐) − 𝜑(𝑡1 , 𝜐)] 𝑑𝜎𝑘 (𝜐)𝐹𝑘 (𝜐, 𝑡2 ) 2 𝜕Ω22 1 1 + 𝜑 (𝑡1 , 𝑡2 ) − 𝜑 (𝑡1󸀠 , 𝑡2󸀠 ) , 4 4

(41)

According to (33), we have 𝑋1 (𝜕Ω12 × 𝜕Ω22 ) − 𝑋1 (𝜕Ω12 × 𝜕Ω22 ) = 𝜆∫

𝜕Ω12 ×𝜕Ω22

[𝐸𝑚 (𝑢, 𝑡1 ) − 𝐸𝑚 (𝑚, 𝑡1󸀠 )] 𝑑𝜎𝑚 (𝑢) 𝜓10 (𝑡1 , 𝑡2 )

× 𝑑𝜎𝑘 (𝜐) 𝐸𝑘 (𝜐, 𝑡2 ) + 𝜆∫

𝜕Ω12 ×𝜕Ω22

󵄩 󵄩 󵄨 󵄨 󵄨󵄨 󸀠 󸀠 󵄨𝛽 󵄨󵄨𝑅 (𝜕Ω12 × 𝜕Ω22 )󵄨󵄨󵄨 ≤ 𝐶4 󵄩󵄩󵄩𝜑󵄩󵄩󵄩𝛽 󵄨󵄨󵄨󵄨(𝑡1 , 𝑡2 ) − (𝑡1 , 𝑡2 )󵄨󵄨󵄨󵄨 .

𝐸𝑚 (𝑢, 𝑡1󸀠 ) 𝑑𝜎𝑚 (𝑢) 𝜓10 (𝑡1 , 𝑡2 ) 𝑑𝜎𝑘 (𝜐)

+ 𝜆∫

𝜕Ω12 ×𝜕Ω22

× ×

𝑑𝜎𝑘 (𝜐) 𝐸𝑘 (𝜐, 𝑡2󸀠 )

(𝑡1 , 𝑡2 ) −

󵄨󵄨 󵄩 󵄩 󵄨 󵄨 󸀠 󸀠 󵄨𝛽 󵄨󵄨𝑅 (𝜕Ω1 × 𝜕Ω2 )󵄨󵄨󵄨 ≤ 𝐶5 󵄩󵄩󵄩𝜑󵄩󵄩󵄩𝛽 󵄨󵄨󵄨󵄨(𝑡1 , 𝑡2 ) − (𝑡1 , 𝑡2 )󵄨󵄨󵄨󵄨 .

(𝑢)

[𝜓10

𝜓10

≤

(45)

Thus we infer

× [𝐸𝑘 (𝜐, 𝑡2 ) − 𝐸𝑘 (𝜐, 𝑡2󸀠 )] 𝐸𝑚 (𝑢, 𝑡1󸀠 ) 𝑑𝜎𝑚

by (35)–(38), we have |𝐸3 + 𝐹3 + 𝐺3 + 𝐻3 | 𝛽 𝐶3 ‖𝜑‖𝛽 |(𝑡1 , 𝑡2 ) − (𝑡1󸀠 , 𝑡2󸀠 )| . Summarizing the above discussion shows that

(44)

(46)

Hence

(𝑡1󸀠 , 𝑡2󸀠 )]

󵄩󵄩 󵄩󵄩 4 󵄩 󵄩󵄩 󵄩󵄩∑𝑋𝑖 (𝑡1 , 𝑡2 )󵄩󵄩󵄩 ≤ 𝐽4 󵄩󵄩󵄩𝜑󵄩󵄩󵄩 . 󵄩󵄩 󵄩󵄩 󵄩 󵄩𝛽 󵄩󵄩𝛽 󵄩󵄩𝑖=1

= 𝐸1 + 𝐸2 + 𝐸3 . (42)

This completes the proof.

(47)


7 where

Corollary 14. If 𝜑(𝑡1 , 𝑡2 ) ∈ 𝐻(𝜕Ω1 × 𝜕Ω2 , 𝛽), then 󵄩󵄩 ++ 󵄩 󵄩 󵄩 󵄩󵄩𝜙 (𝑡1 , 𝑡2 )󵄩󵄩󵄩𝛽 ≤ 𝐽5 󵄩󵄩󵄩𝜑󵄩󵄩󵄩𝛽 , 󵄩 󵄩 󵄩 󵄩󵄩 −+ 󵄩󵄩𝜙 (𝑡1 , 𝑡2 )󵄩󵄩󵄩𝛽 ≤ 𝐽5 󵄩󵄩󵄩𝜑󵄩󵄩󵄩𝛽 ,

𝐴 1 (𝜕Ω1 × 𝜕Ω2 )

󵄩 󵄩 󵄩󵄩 +− 󵄩 󵄩󵄩𝜙 (𝑡1 , 𝑡2 )󵄩󵄩󵄩𝛽 ≤ 𝐽5 󵄩󵄩󵄩𝜑󵄩󵄩󵄩𝛽 , 󵄩 󵄩 󵄩󵄩 −− 󵄩 󵄩󵄩𝜙 (𝑡1 , 𝑡2 )󵄩󵄩󵄩𝛽 ≤ 𝐽5 󵄩󵄩󵄩𝜑󵄩󵄩󵄩𝛽 .

= 4𝜆 ∫

× 𝐸𝑘 (𝜐, 𝑡2 ) ,

Theorem 15. Let 𝐴(𝑥, 𝑦), 𝐵(𝑥, 𝑦), 𝐶(𝑥, 𝑦), 𝐷(𝑥, 𝑦), 𝑔(𝑥, 𝑦) ∈ 𝐻(𝜕Ω1 × 𝜕Ω2 , 𝛽); then the function 𝑓(𝑡1 , 𝑡2 , 𝜙1 , 𝜙2 , 𝜙3 , 𝜙4 ) is a Hölder continuous function for (𝑡1 , 𝑡2 ) ∈ 𝜕Ω1 × 𝜕Ω2 and satisfies the Lipschitz-condition for 𝜙1 , 𝜙2 , 𝜙3 , 𝜙4 and any (𝑡1 , 𝑡2 ), namely,

𝐴 2 (𝜕Ω1 × 𝜕Ω2 ) = −4𝜆 ∫

𝜕Ω1 ×𝜕Ω2

𝐸𝑚 (𝑢, 𝑡1 ) 𝑑𝜎𝑚 (𝑢) 𝜓1 (𝑢, 𝜐) 𝑑𝜎𝑘 (𝜐)

× 𝐸𝑘 (𝜐, 𝑡2 ) , 𝐵1 (𝜕Ω1 × 𝜕Ω2 )

󵄨󵄨 󵄨 󵄨󵄨𝑓 (𝑡11 , 𝑡21 , 𝜙11 , 𝜙12 , 𝜙13 , 𝜙14 ) − 𝑓 (𝑡12 , 𝑡22 , 𝜙21 , 𝜙22 , 𝜙23 , 𝜙24 )󵄨󵄨󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨𝛽 ≤ 𝐽6 󵄨󵄨󵄨(𝑡11 , 𝑡21 ) − (𝑡12 , 𝑡22 )󵄨󵄨󵄨 + 𝐽7 󵄨󵄨󵄨󵄨𝜙11 − 𝜙21 󵄨󵄨󵄨󵄨

𝐸𝑚 (𝑢, 𝑡1 ) 𝑑𝜎𝑚 (𝑢) 𝜓1𝑛 (𝑢, 𝜐) 𝑑𝜎𝑘 (𝜐)

𝜕Ω1 ×𝜕Ω2

(48)

(49)

= −4𝜆 ∫

𝜕Ω1 ×𝜕Ω2

̃ 𝐸𝑚 (𝑢, 𝑡1 ) 𝑑𝜎𝑚 (𝑢) 𝜓2𝑛 (𝑢, 𝜐) 𝑑𝜎 𝑘 (𝜐)

× 𝐹𝑘 (𝜐, 𝑡2 ) ,

󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 + 𝐽8 󵄨󵄨󵄨󵄨𝜙12 − 𝜙22 󵄨󵄨󵄨󵄨 + 𝐽9 󵄨󵄨󵄨󵄨𝜙13 − 𝜙23 󵄨󵄨󵄨󵄨 + 𝐽10 󵄨󵄨󵄨󵄨𝜙14 − 𝜙24 󵄨󵄨󵄨󵄨 ,

𝐵2 (𝜕Ω1 × 𝜕Ω2 )

where 𝐽𝑖 (𝑖 = 6, . . . , 10) is a positive constant and has nothing to do with 𝑡1𝑗 , 𝑡2𝑗 , 𝜙𝑗1 , . . . , 𝜙𝑗4 , 𝑗 = 1, 2. If 𝑓(0, 0, 0, 0, 0, 0) = 0, ‖𝐴 + 𝐵‖𝛽 < 𝜀, ‖𝐶 + 𝐷‖𝛽 < 𝜀, ‖𝐷 + 𝐵‖𝛽 < 𝜀, ‖1 − 4𝐵‖𝛽 < 𝜀, 0 < 𝜀 < 1, 0 < 𝜇 = 𝜀𝐽0 (2𝐽5 + 𝐽2 + 1) < 1, ‖𝑔‖𝛽 < 𝛿, 0 < 𝛿 < 𝑀(1 − 𝜇)/4 ⋅ 𝐽0 (𝐽13 + 𝐽14 𝑀), then the problem R has at least one solution, where 𝑀(‖𝜑‖𝛽 < 𝑀), 𝐽13 , 𝐽14 are both positive constants satisfying ‖𝑓‖𝛽 ≤ 𝐽13 + 𝐽14 ‖𝜑‖𝛽 . Proof. Suppose that 𝑇 = {𝜑 | 𝜑 ∈ 𝐻(𝜕Ω1 × 𝜕Ω2 , 𝛽), ‖𝜑‖𝛽 < 𝑀} be denoted a subset of 𝐶(𝜕Ω1 × 𝜕Ω2 ). From Theorem 13, Corollary 14, and (38), we obtain 𝐶(𝑓, 𝜕Ω1 × 𝜕Ω2 ) ≤ 𝐽11 + 𝐽12 ‖𝜑‖𝛽 . Similarly, we can get ‖𝑓‖𝛽 ≤ 𝐽13 + 𝐽14 ‖𝜑‖𝛽 . Hence, by (12) and 𝐹𝜑 = 𝜑, ‖𝐹𝜑‖𝛽 ≤ 𝑀 is derived. This shows that the operator 𝐹 is the mapping of 𝑇 → 𝑇. Next we prove the 𝐹 is a continuous mapping. Suppose that the sequence of functions {𝜑𝑛 } ∈ 𝑇 uniformly converges to a function 𝜑(𝑡1 , 𝑡2 ), (𝑡1 , 𝑡2 ) ∈ 𝜕Ω1 × 𝜕Ω2 ; thus for arbitrary 𝜀 > 0 and if 𝑛 is large enough, then |(𝑃𝑖 + 𝑄𝑖 )𝜑𝑛 − (𝑃𝑖 + 𝑄𝑖 )𝜑| < 𝜀, (𝑖 = 1, 2). Now we consider 𝑃3 𝜑𝑛 − 𝑃3 𝜑 by

̃ 𝐸𝑚 (𝑢, 𝑡1 ) 𝑑𝜎𝑚 (𝑢) 𝜓2 (𝑢, 𝜐) 𝑑𝜎 𝑘 (𝜐)

= 4𝜆 ∫

𝜕Ω1 ×𝜕Ω2

× 𝐹𝑘 (𝜐, 𝑡2 ) 𝐶1 (𝜕Ω1 × 𝜕Ω2 ) = −4𝜆 ∫

𝜕Ω1 ×𝜕Ω2

̂ 𝐹𝑚 (𝑢, 𝑡1 ) 𝑑𝜎 𝑚 (𝑢)𝜓3𝑛 (𝑢, 𝜐) 𝑑𝜎𝑘 (𝜐)

× 𝐸𝑘 (𝜐, 𝑡2 ) , 𝐶2 (𝜕Ω1 × 𝜕Ω2 ) = 4𝜆 ∫

𝜕Ω1 ×𝜕Ω2

̂ 𝐹𝑚 (𝑢, 𝑡1 ) 𝑑𝜎 𝑚 (𝑢)𝜓3 (𝑢, 𝜐) 𝑑𝜎𝑘 (𝜐) × 𝐸𝑘 (𝜐, 𝑡2 ) ,

𝐷1 (𝜕Ω1 × 𝜕Ω2 ) = 4𝜆 ∫

𝜕Ω1 ×𝜕Ω2

̃ ̂ 𝐹𝑚 (𝑢, 𝑡1 ) 𝑑𝜎 𝑚 (𝑢)𝜓4𝑛 𝑑𝜎𝑘 (𝜐)𝐹𝑘 (𝜐, 𝑡2 ) ,

𝐷2 (𝜕Ω1 × 𝜕Ω2 ) 𝑃3 𝜑𝑛 − 𝑃3 𝜑

= −4𝜆 ∫

𝜕Ω1 ×𝜕Ω2

2

2

𝑖=1

𝑖=1

= ∑𝐴 𝑖 (𝜕Ω1 × 𝜕Ω2 ) + ∑𝐵𝑖 (𝜕Ω1 × 𝜕Ω2 )

̃ ̂ 𝐹𝑚 (𝑢, 𝑡1 ) 𝑑𝜎 𝑚 (𝑢)𝜓4 𝑑𝜎𝑘 (𝜐)𝐹𝑘 (𝜐, 𝑡2 ) ,

𝐸1 = 𝜑 (𝑡1 , 𝑡2 ) − 𝜑𝑛 (𝑡1 , 𝑡2 ) ,

2

2

3

𝐸2 = (𝑃1 + 𝑄1 ) 𝜑𝑛 − (𝑃1 + 𝑄1 ) 𝜑,

𝑖=1

𝑖=1

𝑖=1

𝐸3 = (𝑃2 + 𝑄2 ) 𝜑𝑛 − (𝑃2 + 𝑄2 ) 𝜑.

+ ∑𝐶𝑖 (𝜕Ω1 × 𝜕Ω2 ) + ∑𝐷𝑖 (𝜕Ω1 × 𝜕Ω2 ) + ∑𝐸𝑖 , (50)

(51)

8


Suppose 6𝛿 < 𝑑𝑖 , 𝑖 = 1, 2, 𝛿 > 0, 𝑂((𝑡1 , 𝑡2 ), 3𝛿) is the 3𝛿-neighborhood of (𝑡1 , 𝑡2 ) with the center at point (𝑡1 , 𝑡2 ) ∈ 𝜕Ω1 × 𝜕Ω2 and the radius 3𝛿, 𝜕Ω𝑖1 × 𝜕Ω𝑖2 is as above; then r𝑗 (𝜕Ω1 × 𝜕Ω2 ) = r𝑗 (𝜕Ω11 × 𝜕Ω21 ) + r𝑗 (𝜕Ω11 × 𝜕Ω22 ) + r𝑗 (𝜕Ω12 × 𝜕Ω21 ) + r𝑗 (𝜕Ω12 × 𝜕Ω22 ) ,

(52)

󵄨󵄨 󵄨 𝛽 𝛽/2 󵄨󵄨𝐴 1 (𝜕Ω11 × 𝜕Ω21 )󵄨󵄨󵄨 ≤ 𝐽15 𝛿 ≤ 𝐽16 𝛿 , (53)

Similarly, we can get the inequality estimations for 𝐴 2 (𝜕Ω11 × 𝜕Ω21 ), 𝐴 2 (𝜕Ω12 × 𝜕Ω21 ), and 𝐴 2 (𝜕Ω11 × 𝜕Ω22 ). By (35), (36), and (37), we can obtain the similar inequality estimations for 𝐵𝑖 (𝜕Ω11 × 𝜕Ω21 ), 𝐵𝑖 (𝜕Ω12 × 𝜕Ω21 ), 𝐵𝑖 (𝜕Ω11 × 𝜕Ω22 ), 𝐶𝑖 (𝜕Ω11 × 𝜕Ω21 ), 𝐶𝑖 (𝜕Ω12 × 𝜕Ω21 ), 𝐶𝑖 (𝜕Ω11 × 𝜕Ω22 ), 𝐷𝑖 (𝜕Ω11 × 𝜕Ω21 ), 𝐷𝑖 (𝜕Ω12 × 𝜕Ω21 ), and 𝐷𝑖 (𝜕Ω11 × 𝜕Ω22 ), 𝑖 = 1, 2, respectively. From 𝐴 1 (𝜕Ω12 × 𝜕Ω22 ) + 𝐴 2 (𝜕Ω12 × 𝜕Ω22 ) 𝜕Ω12 ×𝜕Ω22

̃ ̃ ̃ ̃ ̂ = {[𝜑̂ 𝑛 (𝑢, 𝜐) − 𝜑 (𝑢, 𝜐)] − [𝜑𝑛 (𝑡1 , 𝜐) − 𝜑 (𝑡1 , 𝜐)]} + {[𝜑𝑛 (𝑡1 , 𝑡2 ) − 𝜑 (𝑡1 , 𝑡2 )] − [𝜑𝑛̂ (𝑢, 𝑡2 ) − 𝜑̂ (𝑢, 𝑡2 )]} , (58) 󵄩 󵄨 󵄩 󵄨󵄨 󵄨󵄨𝐷1 (𝜕Ω12 × 𝜕Ω22 ) + 𝐷2 (𝜕Ω12 × 𝜕Ω22 )󵄨󵄨󵄨 ≤ 𝐽21 󵄩󵄩󵄩𝜑𝑛 − 𝜑󵄩󵄩󵄩𝛽 . (59)

By (35), we can obtain that

= 4𝜆 ∫

𝑊4 (𝑢, 𝜐)

since |𝑊4 (𝑢, 𝜐)| ≤ 4‖𝜑𝑛 − 𝜑‖𝛽 and from (38), we obtain that

(r = 𝐴, 𝐵, 𝐶, 𝐷, 𝑗 = 1, 2) .

󵄨 󵄨󵄨 𝛽/2 󵄨󵄨𝐴 1 (𝜕Ω12 × 𝜕Ω21 )󵄨󵄨󵄨 ≤ 𝐽17 𝛿 , 󵄨 󵄨󵄨 𝛽/2 󵄨󵄨𝐴 1 (𝜕Ω11 × 𝜕Ω22 )󵄨󵄨󵄨 ≤ 𝐽18 𝛿 .

where

𝐸𝑚 (𝑢, 𝑡1 ) 𝑑𝜎𝑚 (𝑢) 𝑊1 (𝑢, 𝜐) 𝑑𝜎𝑘 (𝜐) (54)

× 𝐸𝑘 (𝜐, 𝑡2 ) ,

Summarizing the above discussion, we conclude |𝑃3 𝜑𝑛 − 𝑃3 𝜑| ≤ 𝐽22 (𝜀 + 𝛿𝛽/2 + ‖𝜑𝑛 − 𝜑‖𝛽 ). Then for arbitrary 𝜀 > 0, we first choose a sufficiently small number 𝛿 and next select a sufficiently large positive integer 𝑛; we have 󵄨󵄨󵄨𝑃3 𝜑𝑛 − 𝑃3 𝜑󵄨󵄨󵄨 < 𝐺𝜀, (60) 󵄨 󵄨 where 𝐺 is a positive constant. Finally, we can choose 𝑛 large enough such that |𝐹𝜑𝑛 − 𝐹𝜑| < 𝑊𝜀 (𝑊 is a positive constant). Hence we can obtain 𝐹 : 𝑇 → 𝑇 is a continuous mapping. According to Ascoli-Arzela Theorem, 𝑇 is a compact set in the space 𝐶(𝜕Ω1 ×𝜕Ω2 ). Based on the Schauder fixed point principle, there exists a function 𝜑 ∈ 𝐻(𝜕Ω1 × 𝜕Ω2 , 𝛽) satisfying the equation 𝐹𝜑 = 𝜑. Corollary 16. If 𝑓 ≡ 1 in Theorem 15, then the Problem R has the unique solution. Proof. This corollary is not difficult to verify by the contraction mapping principle when 𝑓 ≡ 1 in Theorem 15.

Conflict of Interests

where

The authors declare that there is no conflict of interests regarding the publication of this paper.

𝑊1 (𝑢, 𝜐) = {[𝜑𝑛 (𝑢, 𝜐) − 𝜑 (𝑢, 𝜐)] − [𝜑𝑛 (𝑡1 , 𝜐) − 𝜑 (𝑡1 , 𝜐)]} + {[𝜑𝑛 (𝑡1 , 𝑡2 ) − 𝜑 (𝑡1 , 𝑡2 )] − [ 𝜑𝑛 (𝑢, 𝑡2 ) − 𝜑 (𝑢, 𝑡2 )} , (55) since |𝑊1 (𝑢, 𝜐)| ≤ 2‖𝜑𝑛 − 𝜑‖𝛽 |𝑢 − 𝑡1 |𝛽/2 |𝜐 − 𝑡2 |𝛽/2 and from (35), we have 󵄨󵄨 󵄩 󵄩 󵄨 󵄨󵄨𝐴 1 (𝜕Ω12 × 𝜕Ω22 ) + 𝐴 2 (𝜕Ω12 × 𝜕Ω22 )󵄨󵄨󵄨 ≤ 𝐽19 󵄩󵄩󵄩𝜑𝑛 − 𝜑󵄩󵄩󵄩𝛽 . (56) Similarly, we can get the inequality estimations for 𝐵1 (𝜕Ω12 × 𝜕Ω22 ) + 𝐵2 (𝜕Ω12 × 𝜕Ω22 ),𝐶1 (𝜕Ω12 × 𝜕Ω22 ) + 𝐶2 (𝜕Ω12 × 𝜕Ω22 ). From 𝐷1 (𝜕Ω12 × 𝜕Ω22 ) + 𝐷2 (𝜕Ω12 × 𝜕Ω22 ) = 4𝜆 ∫

𝜕Ω12 ×𝜕Ω22

̃ ̂ 𝐹𝑚 (𝑢, 𝑡2 ) 𝑑𝜎 𝑚 (𝑢)𝑊4 (𝑢, 𝜐) 𝑑𝜎𝑘 (𝜐)𝐹𝑘 (𝜐, 𝑡2 ) , (57)

Acknowledgment This work was completed with the support of the National Natural Science Foundation of China (NNSFC) through Grants no. 10671207 and no. 11301136.

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