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The mixed boundary value problem for the inhomogeneous Cimmino system
Boundary Value Problems volume 2015, Article number: 13 (2015)
Abstract
In this article, we first propose a kind of mixed boundary value problem for the inhomogeneous Cimmino system, which consists of first order linear partial differential equations in \(\mathbb{R}^{4}\). Then, by using the one-to-one correspondence between the theory of quaternion valued hyperholomorphic functions and that of Cimmino system’s solutions, we transform the problem as stated above into a problem related to the ψ-hyperholomorphic functions in quaternionic analysis. Moreover, we show the boundedness, Hölder continuity, and generalized derivatives of a kind of singular integral operator \({}^{\psi } T_{\mathbb{C}^{2}}[g]\) related to ψ-hyperholomorphic functions in quaternionic analysis. Lastly, the solution of the mixed boundary value problem for the inhomogeneous Cimmino system is explicitly described.
1 Introduction
The skew field of quaternions ℍ gives an example of a noncommutative Clifford algebra with minimal dimension. It serves as a very convenient model of general Clifford constructions. Today, quaternionic analysis is regarded as a broadly accepted branch of classical analysis offering a successful generalization of complex analysis. It studies functions defined on domains in \(\mathbb{R}^{3}\) or \(\mathbb{R}^{4}\) with values in the skew field of real quaternions ℍ. This theory is centered around the concept of ψ-hyperholomorphic functions related to a so-called structural set ψ of \(\mathbb{H}^{3}\) or \(\mathbb{H}^{4}\), respectively.
Quaternionic analysis initiated new solution methods for boundary value problems in several research areas of mathematical physics, in particular in planar fluids, quantum field theory, electromagnetic wave equations etc. Many scholars and experts have studied some boundary and initial value problems in higher dimensions by using them, such as Gürlebeck, Sprössig, Adler, Alesker, Yang, and so on [1–5].
The Cimmino system (1.1) offers a natural and elegant generalization to the four-dimensional case of that of Cauchy-Riemann. Cimmino, Dragomir and Lanconelli have done a lot of research on it [6, 7]. Recently, Abreu Blaya et al. [8] studied the Dirichlet boundary value problem for the inhomogeneous Cimmino system (1.2). We have
where \(f_{m}\) (\(m=0,1,2,3\)) are continuously differentiable ℝ-valued functions in \(\Omega\subset\mathbb{R}^{4}\). The corresponding inhomogeneous Cimmino system is as follows:
where \(f_{m}\) are as stated above, \(g_{m}\in L_{p}(\Omega, \mathbb {R})\) (\(m=0,1,2,3\)).
In this article, we will study a kind of mixed boundary value problem for the inhomogeneous Cimmino system (1.2) by using the quaternionic analysis approach. This article is organized as follows. In Section 2, we recall some basic knowledge of quaternionic analysis. In Section 3, we construct a singular integral operator and study some of its properties. In Section 4, we first propose a kind of mixed boundary value problem for the inhomogeneous Cimmino system (1.2); then we obtain an integral representation of the solution of the mixed boundary value problem by using the one-to-one correspondence between the theory of quaternion valued hyperholomorphic functions and that of a Cimmino system’s solutions.
2 Preliminaries
Quaternionic analysis studies functions defined on \(\mathbb{R}^{4}\) with their values in quaternion algebra space ℍ, which is a four-dimensional vector space with basis e, i, j, k. The basis element e is a unit element, henceforth we shall abbreviate e to 1. Also, i, j, k satisfy the following multiplication rule:
An arbitrary element of the quaternion algebra space ℍ can be written as \(x=x_{0}+ix_{1}+jx_{2}+kx_{3}\), \(x_{m}\in\mathbb{R}\) (\(m=0,1,2,3\)), and \(\bar {x}=x_{0}-ix_{1}-jx_{2}-kx_{3}\). The norm for an element \(x\in\mathbb{H}\) is taken to be \(|x|=\sqrt{x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}}\) and satisfies \(|\bar{x}|=|x|\), \(|x+y|\leq{|x|+|y|}\), \(|xy|=|x||y|\). Obviously, \(\overline{xy}=\bar{y}\bar{x}\) and \(x\bar{x}=\bar{x}x=|x|^{2}\). In addition, suppose the imaginary unit of ℂ is identified with the basis element i in quaternion algebra space ℍ, then for arbitrary \(z\in\mathbb{C}\), we have \(z=x_{0}+ix_{1}\) and its complex conjugate \(\bar{z}=x_{0}-ix_{1}\). In this way it is easily seen that \(zj=j\bar{z}\).
By means of the mapping \(x_{0}+ix_{1}+jx_{2}+kx_{3}\rightarrow (x_{0}+ix_{1})+(x_{2}+ix_{3})j\) (\(\rightarrow(x_{0},x_{1},x_{2},x_{3})\)), one can see ℍ as \(\mathbb{C}^{2}\) (or \(\mathbb{R}^{4}\)). From now on, an arbitrary element \(\xi\in\mathbb{H}\) can be written as \(\xi =z_{1}+z_{2}j\), \(z_{1},z_{2}\in\mathbb{C}\). From the multiplication rule as stated above, for arbitrary \(\xi=z_{1}+z_{2}j\), \(\eta=\varsigma _{1}+\varsigma_{2}j\in\mathbb{H}\), \(z_{1},z_{2},\varsigma_{1},\varsigma _{2}\in\mathbb{C}\). We have \(\xi\eta=(z_{1}\varsigma_{1}-z_{2}\bar{\varsigma} _{2})+(z_{1}\varsigma_{2}+z_{2}\bar{\varsigma}_{1})j\), \(\bar{\xi }=\overline{z_{1}+z_{2}j}=\bar{z}_{1}+\overline{z_{2}j}=\bar {z}_{1}-z_{2}j\), and \(\xi\bar{\xi}=\bar{\xi}\xi =|z_{1}|^{2}+|z_{2}|^{2}=|\xi|^{2}\).
Let \(\Omega\subset\mathbb{R}^{4}\) be a nonempty open bounded connected set and the boundary \(\Gamma=\partial{\Omega}\) be a differentiable, oriented, and compact Liapunov surface. The functions f which are defined in Ω with values in ℍ can be expressed as \(f(x)=f_{0}+f_{1}i+f_{2}j+f_{3}k\), where \(f_{m}\) (\(m=0,1,2,3\)) are continuously differentiable ℝ-valued functions in \(\Omega \subset\mathbb{R}^{4}\). On \(C^{(1)}(\Omega,\mathbb{H})\), we define the differential operators ψ D and \({}^{\bar{\psi}} D\) as follows:
where
Obviously, the differential operators ψ D and \({}^{\bar{\psi}} D\) can be written as
which are associated to the structural set \(\psi=\{1,i,-j,k\}\) and \(\bar {\psi}=\{1,-i,j,-k\}\), respectively.
Let \(\Delta_{\mathbb{R}^{4}}=\sum_{m=0}^{3}\partial_{x_{m}}^{2}\), then the following equalities hold on \(C^{(2)}(\Omega,\mathbb{H})\):
Taking into account that the multiplication in ℍ is noncommutative, the functions f are called left ψ-hyperholomorphic in Ω if \({}^{\psi} D[f](\xi)=0\) (\(\xi\in\Omega\)). The functions g are called right ψ-hyperholomorphic in Ω if \([g]{{}^{\psi} D}(\xi )=0\) (\(\xi\in\Omega\)).
Denote by \(\Theta_{4}\) the fundamental solution of the Laplace operator
and by \(\mathcal{K}_{\psi}\) the fundamental solution of the operator ψ D:
Then the corresponding Cauchy type integral operator is
and the Teodorescu type integral operator is
In this article, \(g(x)\in L_{p}(\mathbb{C}^{2},\mathbb{H})\) means that \(g(x)\in L_{p}(E,\mathbb{H})\), \(g_{\sigma}(x)=|x|^{-\sigma}g(\frac{\bar {x}}{|x|^{2}})\in L_{p}(E,\mathbb{H})\), in which \(E=\{\xi||\xi|\leq1\} \), σ is a real number, \(\| g\|_{L_{p}}=\|g\|_{L_{p}(E)}+\| g_{\sigma}\|_{L_{p}(E)}\), \(p\geq1\). The following fundamental statements are widely known to hold and can be found in [1, 9, 10], respectively.
Definition 2.1
Suppose that the functions f, g, φ are defined in Ω with values in ℍ and \(f,g\in L_{1}(\Omega,\mathbb {H})\). If for arbitrary \(\varphi\in C_{0}^{\infty}(\Omega,\mathbb{H})\), f, g satisfy
then f is called a generalized derivative of the function g, denoted by \(f={{}^{\psi} D[g]}\).
Lemma 2.1
([9])
If \(\sigma_{1},\sigma_{2}>0\), \(0\leq\gamma\leq1\), then we have
Lemma 2.2
(Integral form of the quaternionic Stokes formula [1])
Let \(\Omega,\Gamma=\partial\Omega\) be as stated above and \(f, g\in C^{(1)}(\Omega,\mathbb{H})\), then
Lemma 2.3
(Borel-Pompeiu quaternionic formula [1])
Let \(\Omega ,\Gamma=\partial\Omega\) be as stated above and \(f\in C^{(1)}(\Omega ,\mathbb{H})\), then for arbitrary \(\xi\in\Omega\), we have
and
Lemma 2.4
(Hadamard lemma [10])
Suppose Ω be as stated above. If \(\alpha'\), \(\beta'\) satisfy \(0<\alpha', \beta'<4\), \(\alpha'+\beta '>4\), then for all \(x_{1},x_{2}\in\mathbb{R}^{4}\) and \(x_{1}\neq x_{2}\), we have
3 Some useful properties of the Cimmino singular integral operator
By means of the idea as stated above, we suppose
Then system (1.1) can be written as
Moreover, if the pair \((u_{1},u_{2})\) of continuously differentiable (up to the second order) complex-valued functions give a solution of system (1.1) then
A similar observation is valid for \(\bar{u}_{2}\), so \(u_{1}\), \(\bar {u}_{2}\) are complex-valued harmonic functions, i.e. the set of solutions of system (1.1) contains all holomorphic functions of two complex variables.
Similarly, system (1.2) can be written as
The generalized Teodorescu type integral operator \({}^{\psi} T_{\mathbb {C}^{2}}[g]\) can be written as
where the Cimmino singular integral operators \({}^{\psi} T_{\mathbb {C}^{2}}^{(1)}[g]\), \({{}^{\psi} T_{\mathbb{C}^{2}}^{(2)}[g]}\) are as follows:
Theorem 3.1
Let E be as stated above. If \(g\in L_{p}(\mathbb {C}^{2},\mathbb{H})\), \(4< p<+\infty\), then we have
-
(1)
\(|{}^{\psi} T_{\mathbb{C}^{2}}[g](\xi)|\leq M_{1}(p)\|g\|_{L_{p}}\), \(\xi \in\mathbb{C}^{2}\cong\mathbb{R}^{4}\),
-
(2)
\({}^{\psi} T_{\mathbb{C}^{2}}[g]\in C_{\beta}(\mathbb{C}^{2},\mathbb {H})\cong C_{\beta}(\mathbb{R}^{4},\mathbb{H})\) (\(0<\beta=1-4/p<1\)),
-
(3)
\({}^{\psi} D({}^{\psi} T_{\mathbb{C}^{2}}[g])(\xi)=g(\xi)\), \(\xi\in\mathbb {C}^{2}\cong\mathbb{R}^{4}\).
Proof
(1) First, we have
By the Hölder inequality, we have
where \(1/p+1/q=1\).
When \(\xi\in\bar{E}\), because of \(4< p<+\infty\), \(1/p+1/q=1\), we have \(1< q<4/3\). Thus \(\int_{E}\frac{1}{|\xi-\eta|^{3q}}\, dE_{\eta}\) is bounded. Hence we have
When \(\xi\in\mathbb{C}^{2}-\bar{E}\), by Lemma 2.1 and the generalized spherical coordinate, we have
where \(\rho=|\xi-\eta|\), \(d_{0}=d(\xi, \bar{E})\).
Therefore, for \(\forall\xi\in\mathbb{C}^{2}\cong\mathbb{R}^{4}\), we can obtain
where \(M'_{1}(p)=\max\{J_{1}/\pi^{2},J_{3}/\pi^{2}\}\).
For \(\eta\in\mathbb{C}^{2}-E\), we suppose that \(\eta=\frac{\bar {\eta}'}{|\eta'|^{2}}\), then we have \(|\eta'|\leq1\). Thus by \(g\in L_{p}(\mathbb{C}^{2},\mathbb{H})\), similar to the proof as stated above, we have
Therefore, we obtain
where \(M_{1}(p)=M'_{1}(p)+M''_{1}(p)\).
(2) For arbitrary \(\xi',\xi''\in\mathbb{C}^{2}\cong\mathbb{R}^{4}\), \(\xi '\neq\xi''\), we have
and
Since
Thus
Again, because of
Thus by (3.5), (3.6), and the Hölder inequality, we have
Suppose \(\alpha_{l}=(4-l)q\), \(\beta_{l}=lq\) (\(l=1,2,3\)). By \(1< q<4/3\), we know
Thus, by Lemma 2.4, for \(l=1,2,3\), we have
Thus, by inequalities (3.7) and (3.8), we obtain
where \(0<\beta=1+(4-4q)/q=1-4/p<1\).
Next, we discuss \(I_{1_{2}}\).
For arbitrary \(\xi',\xi''\in\mathbb{C}^{2}\cong\mathbb{R}^{4}\), \(\xi'\neq \xi''\), we suppose \(|\xi'-\xi''|=\delta\) and construct a sphere \(B(\xi ',3\delta)\) with the center at \(\xi'\) and radius 3δ. Next we discuss \(I_{1_{2}}\) in two cases.
(i) If \(B(\xi',3\delta)\cap\bar{E}\neq\emptyset\), then we may suppose \(B(\xi',3\delta)\cap\bar{E}=\Omega_{1}\), \(\bar{E}-\Omega_{1}=\Omega_{2}\). Thus we have
Again, by inequality (3.4), the Hölder inequality, and the use of a local generalized spherical coordinate, we have
In addition,
Again, because of
Thus, we have
So by (3.10) and (3.12), we obtain
First, similar to the method estimating \(I_{1_{1}}\), we have
Second, when \(\eta\in\Omega_{2}\), \(|\xi'-\eta|>3\delta\), \(|\xi''-\eta |>2\delta\). Thus we have
So we know
Thus, by (3.16), the Hölder inequality, and Lemma 2.4, we can obtain
So, by (3.13), (3.14), and (3.17), we have
Therefore, by (3.10), (3.11), and (3.18), we can obtain
(ii) If \(B(\xi',3\delta)\cap\bar{E}=\emptyset\), then for arbitrary \(\eta\in E\), we have \(|\xi'-\eta|>3\delta\), \(|\xi''-\eta|>2\delta\). Thus similar to the method estimating \(I_{1_{2}}^{(2)}\), we have
So, by (3.19) and (3.20), we have
where \(J_{13}=\max\{J_{11},J_{12}\}\).
Thus, to sum up, by (3.5), (3.9), and (3.21), we obtain
Similarly, we have
So, by (3.3), (3.22), and (3.23), we obtain
where \(M'_{2}(p)=J_{14}+J_{15}\).
For \(\eta\in\mathbb{C}^{2}-E\), we suppose that \(\eta=\frac{\bar {\eta}'}{|\eta'|^{2}}\), then we have \(|\eta'|\leq1\). Thus by \(g\in L_{p}(\mathbb{C}^{2},\mathbb{H})\), similar to the proof as stated above, we have
Therefore, for arbitrary \(\xi',\xi''\in\mathbb{C}^{2}\cong\mathbb {R}^{4}\), \(\xi'\neq\xi''\), we obtain
where \(M_{2}(p)=M'_{2}(p)+M''_{2}(p)\), i.e. \({}^{\psi} T_{\mathbb {C}^{2}}[g]\in C_{\beta}(\mathbb{C}^{2},\mathbb{H})\cong C_{\beta }(\mathbb{R}^{4},\mathbb{H})\) (\(0<\beta<1\)).
(3) For arbitrary \(\varphi\in C_{0}^{\infty}(\mathbb{C}^{2},\mathbb {H})\), there exists a bounded closed set \(Q\subset\mathbb{C}^{2}\), such that \(\overline{\operatorname{supp}\varphi}\subset\subset Q\). Thus, by \(T_{\mathbb {C}^{2}}[g](\infty)=0\), Definition 2.1, Lemma 2.3, and the Fubini theorem, we have
where \(d=\sup_{\xi',\xi''\in Q}|\xi'-\xi''|\). Hence, in the sense of generalized derivatives, we have \({}^{\psi} D({}^{\psi} T_{\mathbb {C}^{2}}[g])(\xi)=g(\xi)\). □
Remark 3.1
By the process of proof in Theorem 3.1, it is easy to show that \({}^{\psi} T_{\mathbb{C}^{2}}^{(1)}[g], {}^{\psi} T_{\mathbb {C}^{2}}^{(2)}[g]\in C_{\beta}(\mathbb{C}^{2},\mathbb{C})\cong C_{\beta }(\mathbb{R}^{4},\mathbb{C})\) (\(0<\beta<1\)).
4 Integral representation of solution of the mixed boundary value problem for the inhomogeneous Cimmino system
In this section, let \(E=E_{1}\times E_{2}\) be a bounded domain, \(\partial E_{m}\) (\(m=1,2\)) be simply closed curves in the \(z_{m}\)-plane, and \(\partial E_{m}\in C_{\mu}^{(1)}\), \(0<\mu<1\). Without loss of generality, we may consider \(\partial E_{m}=\{z_{m}||z_{m}|=1\}\) and \(E_{m}=\{z_{m}||z_{m}|<1\}\) (\(m=1,2\)). Denote by \(E_{m}^{+}\), \(E_{m}^{-}\) the inner domain and outer domain of \(\partial E_{m}\), respectively, and \(E^{++}=E_{1}^{+}\times E_{2}^{+}\), \(E^{+-}=E_{1}^{+}\times E_{2}^{-}\), \(E^{-+}= E_{1}^{-}\times E_{2}^{+}\), \(E^{--}=E_{1}^{-}\times E_{2}^{-}\), \(\Gamma=\partial E_{1}\times \partial E_{2}\).
Problem P
The mixed boundary value problem for the inhomogeneous Cimmino system (1.2) is to find a function \(f(z_{1},z_{2})=u_{1}(z_{1},z_{2})+u_{2}(z_{1},z_{2})j\) satisfying the Cimmino system (1.2) and the following boundary condition:
where \(u_{1}=f_{0}+if_{1}\), \(u_{2}=f_{2}+if_{3}\), \(z_{1}=x_{0}+ix_{1}\), \(z_{2}=x_{2}+ix_{3}\). \(G_{1}(z_{1},z_{2})\), \(G_{2}(z_{1},z_{2})\), \(G_{3}(z_{1},z_{2})\) are analytic in \(E^{+-}\), \(E^{-+}\), \(E^{--}\) and are continuous in \(\bar {E}^{+-}\), \(\bar{E}^{-+}\), \(\bar{E}^{--}\), respectively, which have no zero. We have \(G_{m}(t_{1},t_{2})\) (\(m=1,2,3\)), \(H(t_{1},t_{2})\in C_{\alpha}(\Gamma,\mathbb{C})\), \(h(t_{1},t_{2})\in C_{\alpha}(\partial E,\mathbb{C})\) (\(0<\alpha<1\)).
Lemma 4.1
If \(\Psi\in C^{(2)}(E,\mathbb{H})\), \(h\in C_{\alpha}(\partial E,\mathbb {C})\) (\(0<\alpha<1\)), \(g\in L_{p}(\mathbb{C}^{2},\mathbb{H})\) (\(4< p<+\infty\)), then the equation \({}^{\psi} D[\Psi]=0\) with the boundary condition \(\bar {w}_{2}|_{\partial E}=\bar{h}(t_{1},t_{2})-\overline{{{}^{\psi} T_{\mathbb {C}^{2}}^{(2)}[g]}}(t_{1},t_{2})\) has the solution \(\Psi =w_{1}+w_{2}j=w_{1}+j\bar{w}_{2}\) and
or
where ν is the unit outward normal on ∂E, \(G(\xi,\eta)\) is the Green’s function in \(E=E_{1}\times E_{2}\), \(\Phi(z_{1},z_{2})\) is an arbitrary analytic function in \(E=E_{1}\times E_{2}\), and
Proof
From Remark 3.1, we know \({}^{\psi} T_{\mathbb {C}^{2}}^{(2)}[g]\in C_{\beta}(\mathbb{C}^{2},\mathbb{C})\cong C_{\beta }(\mathbb{R}^{4},\mathbb{C})\) (\(0<\beta<1\)). Thus by [9], we have \(\bar{h}-\overline{{{}^{\psi} T_{\mathbb {C}^{2}}^{(2)}[g]}}\in C_{\mu}(\partial E,\mathbb{C})\) (\(0<\mu=\min\{\alpha ,\beta\}<1\)). So we may construct
where ν is the unit outward normal on ∂E, \(G(\xi,\eta)\) is the Green’s function in \(E=E_{1}\times E_{2}\), and
Then \(\bar{w}_{2}(\xi)\) is a complex-value harmonic function in E, i.e. \(\Delta_{\mathbb{C}^{2}}\bar{w}_{2}=4(\partial_{z_{1}\bar {z}_{1}}^{2}+\partial_{z_{2}\bar{z}_{2}}^{2})\bar{w}_{2}=0\). Hence
Again, by (3.1), we have
By (4.3), we know \(-\partial_{z_{2}}\bar{w}_{2}\), \(\partial_{z_{1}}\bar {w}_{2}\) satisfy the compatibility condition
Thus by Theorem 7.2.1 of Chapter 7 in [11], the general solution \(w_{1}(z_{1},z_{2})\) of system (4.4) possesses the form
where \(\Phi(z_{1},z_{2})\) is an arbitrary analytic function in \(E=E_{1}\times E_{2}\) and
□
Lemma 4.2
Let \(G_{m}\) (\(m=1,2,3\)), H, \(w_{0}\), E etc. be as stated above. Find a sectionally analytic function \(\Phi(z_{1},z_{2})\) in \(E^{++}\), \(E^{+-}\), \(E^{-+}\), \(E^{--}\), such that \(\Phi(z_{1},z_{2})\) is continuous in \(E^{++}\), \(E^{+-}\), \(E^{-+}\), \(E^{--}\) and satisfies the boundary condition
where
Then the solution has the form
where
and \(\widetilde{H}=(G_{1}+G_{2}+G_{3}-1)(w_{0}+{{}^{\psi} T_{\mathbb {C}^{2}}^{(1)}[g]})+H\).
Proof
From Remark 3.1, we know \({}^{\psi} T_{\mathbb {C}^{2}}^{(1)}[g]\in C_{\beta}(\mathbb{C}^{2},\mathbb{C})\cong C_{\beta }(\mathbb{R}^{4},\mathbb{C})\) (\(0<\beta<1\)). Thus by [9], we have \(\widetilde{H}=(G_{1}+G_{2}+G_{3}-1)(w_{0}+{{}^{\psi } T_{\mathbb{C}^{2}}^{(1)}[g]})+H\in C_{\mu}(\Gamma,\mathbb{C})\) (\(0<\mu =\min\{\alpha,\beta\}<1\)). Hence by Theorem 7.1.2 of Chapter 7 in [11], it is not difficult to verify this lemma. □
Theorem 4.1
Let E, ∂E etc. be as stated above. If \(g\in L_{p}(\mathbb{C}^{2},\mathbb{H})\) (\(4< p<+\infty\)), then the solution of Problem P can be expressed as
where \({}^{\psi} D[\Psi]=0\) and
herein \(w_{0}\), \({{}^{\psi} T_{\mathbb{C}^{2}}^{(2)}[g]}\) are as stated in Lemma 4.1, Φ, \({{}^{\psi} T_{\mathbb{C}^{2}}^{(1)}[g]}\) are as stated in Lemma 4.2.
Proof
By Theorem 3.1, we know \({}^{\psi} D[{{}^{\psi} T_{\mathbb {C}^{2}}[g]}](\xi)=g(\xi)\), thus \({}^{\psi} D[\Psi(\xi)+{{}^{\psi} T_{\mathbb {C}^{2}}[g]}(\xi)]=g(\xi)\). Hence, by (3.2), we know the general solution of system (1.2) has the form
where \({}^{\psi} D[\Psi]=0\), \(\xi=z_{1}+z_{2}j\), \(f(\xi )=f(z_{1},z_{2})=u_{1}(z_{1},z_{2})+u_{2}(z_{1},z_{2})j=u_{1}(z_{1},z_{2})+j\bar {u}_{2}(z_{1},z_{2})\), \(\Psi(\xi)=\Psi (z_{1},z_{2})=w_{1}(z_{1},z_{2})+w_{2}(z_{1},z_{2})j=w_{1}(z_{1},z_{2})+j\bar {w}_{2}(z_{1},z_{2})\), and
Thus
So the boundary condition (4.2) in Problem P can be written as
Therefore, by Lemma 4.1, the solution to the equation \({}^{\psi} D[\Psi ]=0\) with boundary condition (4.8) can be expressed as
where \(w_{1}\), \(\bar{w}_{2}\) are as stated in Lemma 4.1. Again, by (4.7), we have
From Lemma 4.1, we have
where \(\Phi(z_{1},z_{2})\) is an arbitrary analytic function in \(E=E_{1}\times E_{2}\), \(w_{0}\) is as stated in Lemma 4.1. In addition, by Chapter 7 in [11], we know \(w_{0}\in C_{\alpha}(\mathbb {C}^{2},\mathbb{C})\) (\(0<\alpha<1\)), by Remark 3.1, we know \({{}^{\psi } T_{\mathbb{C}^{2}}^{(1)}[g]}\in C_{\beta}(\mathbb{C}^{2},\mathbb {C})\) (\(0<\beta<1\)). So the boundary condition (4.1) in Problem P can be written as
Therefore, by Lemma 4.2, we know \(\Phi(z_{1},z_{2})\) can be expressed as (4.6) in Lemma 4.2. In conclusion, we complete the proof. □
References
Gürlebeck, K, Sprössig, W: Quaternionic and Clifford Calculus for Physicists and Engineers. Wiley, New York (1997)
Adler, SL: Quaternionic quantum field theory. Commun. Math. Phys. 104, 611-656 (1986)
Sprössig, W: Quaternionic operator methods in fluid dynamics. Adv. Appl. Clifford Algebras 18, 963-978 (2008)
Alesker, S: Quaternionic Monge-Ampère equations. J. Geom. Anal. 13, 205-238 (2003)
Yang, PW, Li, D: \(H_{\lambda}\)-Regular vector functions and their boundary value problems. Bound. Value Probl. 2012, 75 (2012)
Cimmino, G: Su alcuni sistemi lineari omogenei di equazioni alle derivate parziali del primo ordine. Rend. Semin. Mat. Univ. Padova 12, 89-113 (1941)
Dragomir, S, Lanconelli, E: On first order linear PDE systems all of whose solutions are harmonic functions. Tsukuba J. Math. 30, 149-170 (2006)
Abreu Blaya, R, Bory Reyes, J, Guzmán Adán, A, Schneider, B: Boundary value problems for the Cimmino system via quaternionic analysis. Appl. Math. Comput. 219, 3872-3881 (2012)
Iftimie, V: Functions hypercomplexes. Bull. Math. Soc. Sci. Math. Roum. 57, 279-332 (1965)
Gilbert, RP, Buchanan, JL: First Order Elliptic Systems: A Function Theoretic Approach. Academic Press, New York (1983)
Wen, GC: Recent Progress in Theory and Applications of Modern Complex Analysis. Science Press, Beijing (2010)
Acknowledgements
This work was supported by the National Science Foundation of China (No. 11401162, No. 11171349, No. 11301136), the Natural Science Foundation of Hebei Province (No. A2015205012, No. A2014205069, No. A2014208158) and Hebei Normal University Dr. Fund (No. L2015B03, No. L2015B04).
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Wang, L., Xu, Z. & Qiao, Y. The mixed boundary value problem for the inhomogeneous Cimmino system. Bound Value Probl 2015, 13 (2015). https://doi.org/10.1186/s13661-014-0273-5
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DOI: https://doi.org/10.1186/s13661-014-0273-5