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A kind of boundary value problem for inhomogeneous partial differential system
Journal of Inequalities and Applications volume 2016, Article number: 180 (2016)
Abstract
In this article, we first define a kind of generalized singular integral operator and discuss its properties. Then we propose a kind of boundary value problem for an inhomogeneous partial differential system in \(R^{4}\). Finally, the integral representation of the solution to a boundary value problem for the inhomogeneous partial differential system is obtained using the above singular integral operator.
1 Introduction
Partial differential equations are encountered in many problems of physics, mechanics, mathematical finance, mathematical biology, and other branches of mathematics [1, 2]. It has been a popular topic since the 1960s. So boundary value problems for partial differential system have always been an important and meaningful topics. There are many scholars who studied on it, such as Keldysh [3], Wen [4, 5], Čanić and Kim [6], Taira [7], and so on. In addition, singular integral operators are the core components of solutions of the boundary value problems for a partial differential system and a degenerate partial differential system. So, for many years, many scholars and experts have studied some properties of all kinds of singular integral operators, and they obtained the integral representations of solutions of some partial differential equations. For example, Vekua [8] first discussed in detail some properties of the Teodorescu operator, and Hile [9] studied some properties of the Teodorescu operator in \(R^{n}\). Then Gilbert et al. [10] and Meng [11] studied its many properties in high dimensional complex space. Gürlebeck and Sprössig [12], and Yang [13] discussed its properties and corresponding boundary value problems in the real quaternion analysis.
In this article, we will study the Riemann boundary value problem for a kind of inhomogeneous partial differential system of first order equations in \(R^{4}\) using the Clifford analysis approach. In Section 2, we recall some basic knowledge of Clifford analysis. In Section 3, we construct a singular integral operator and study some of its properties. In Section 4, we first propose the Riemann boundary value problem for a kind of inhomogeneous partial differential system, then we obtain an integral representation of the solution to the Riemann boundary value problem using the relation between the theory of Clifford-valued generalized holomorphic functions and that of the inhomogeneous partial differential system’s solutions.
2 Preliminaries
Let \({\{e_{0},e_{1},e_{2},e_{3}\}}\) be an orthogonal basis of the Euclidean space \(R^{4}\) and \({\mathit{Cl}}_{0,3}\) be the Clifford algebra with basis
where \(e_{0}\) is the real scalar identity element, \(e_{1}\), \(e_{2}\), \(e_{3}\) satisfy the following multiplication rule:
If we denote \(e_{1}e_{2}=e_{4}\), \(e_{1}e_{3}=e_{5}\), \(e_{2}e_{3}=e_{6}\), \(e_{1}e_{2}e_{3}=e_{7}\), then an arbitrary element of the Clifford algebra space \({\mathit{Cl}}_{0,3}\) can be written as \(a=\sum_{j=0}^{7}a_{j}e_{j}\), \(a_{j}\in R\). The Clifford conjugation is defined by \(\bar{a}=a_{0}-\sum_{j=1}^{6}a_{j}e_{j}+a_{7}e_{7}\). The norm for an element \(a\in{\mathit{Cl}}_{0,3}\) is taken to be \(|a|=\sqrt{\sum_{j=0}^{7}|a_{j}|^{2}}\). Moreover, if \(a\bar {a}=\bar{a}a=|a|^{2}\) and \(|a|\neq0\), then we have
Thus, we say that a is reversible if \(a\bar{a}=\bar{a}a=|a|^{2}\) and \(|a|\neq0\). Obviously, its inverse element can be written as \({a^{-1}=\frac{\bar{a}}{|a|^{2}}}\).
Suppose \(\Omega\subset R^{4}\) is a bounded connected domain and the boundary ∂Ω is a differentiable, oriented, and compact Liapunov surface. An arbitrary element \(x\in\Omega\) is denoted by \(x=x_{0}e_{0}+x_{1}e_{1}+x_{2}e_{2}+x_{3}e_{3}\). The function w which is defined in Ω with values in the Clifford algebra space \({\mathit{Cl}}_{0,3}\) can be expressed as \(w=\sum_{j=0}^{7}w_{j}(x)e_{j}\), herein \(w_{j}(x)\) (\(j=0,\ldots,7\)) are real-functions defined on Ω.
Let \(C^{(m)}(\Omega,{\mathit{Cl}}_{0,3})=\{w|w:\Omega\rightarrow {\mathit{Cl}}_{0,3}, w(x)=\sum_{j=0}^{7}w_{j}(x)e_{j},w_{j}(x)\in C^{(m)}(\Omega,R)\}\). We introduce the generalized Cauchy-Riemann operator on \(C^{(1)}(\Omega,{\mathit{Cl}}_{0,3})\) as follows:
w is called a left (right) Clifford holomorphic function, if \(\partial _{x}w(x)=0\) (\(w(x)\partial_{x}=0\)) in Ω. w is called a left (right) generalized Clifford holomorphic function, if \(\partial _{x}w(x)=c(x)\) (\(w(x)\partial_{x}=c(x)\)) in Ω, herein \(c(x)=\sum_{j=0}^{7}c_{j}(x)e_{j}\). Usually a left Clifford holomorphic function and a left generalized Clifford holomorphic function are called a Clifford holomorphic function and a generalized Clifford holomorphic function for short, respectively. And \(w(x)\in L^{p,\sigma}(R^{4},{\mathit{Cl}}_{0,3})\) means that \(w(x)\in L^{p}(B,{\mathit{Cl}}_{0,3})\), \(w^{(\sigma)}(x)=|x|^{-\sigma}w(\frac{\bar {x}}{|x|^{2}})\in L^{p}(B,{\mathit{Cl}}_{0,3})\), in which \(B=\{x||x|< 1\}\), σ is a real number, \(\| w\|_{p,\sigma}=\|w\|_{L^{p}(B)}+\| w^{(\sigma)}\|_{L^{p}(B)}\), \(p\geq1\).
Definition 2.1
Suppose that the functions u, v, φ are defined in Ω with values in \({\mathit{Cl}}_{0,3}\), and \(u,v\in L^{1}(\Omega ,{\mathit{Cl}}_{0,3})\). If for arbitrary \(\varphi\in C_{0}^{\infty}(\Omega ,{\mathit{Cl}}_{0,3})\), u, v satisfy
then u is called a generalized derivative of the function v, where we denote \(u=\partial_{x}v\).
Lemma 2.1
([14])
Let Ω, ∂Ω be as stated above. If \(f\in C^{(m)}(\overline{\Omega},{\mathit{Cl}}_{0,3})\), then for each \(x\in \Omega\), we have
where \({E(x,y)=\frac{\bar{y}-\bar{x}}{|y-x|^{4}}}\).
Lemma 2.2
([15])
If \(\sigma_{1},\sigma_{2}>0\), \(0\leq\gamma \leq1\), then we have
Lemma 2.3
([16])
Suppose Ω is a bounded domain in \(R^{4}\), and let \(\alpha'\), \(\beta'\) satisfy \(0<\alpha', \beta'<4\), \(\alpha'+\beta'>4\). Then for all \(x_{1},x_{2}\in R^{4}\) and \(x_{1}\neq x_{2}\), we have
3 Some properties of the singular integral operator
In this section, we will discuss some properties of the singular integral operator as follows:
where \(B=\{x||x|< 1\}\).
Theorem 3.1
Assume B to be as stated above. If \(g\in L^{p}(B,{\mathit{Cl}}_{0,3})\), \(4< p<+\infty\), then
-
(1)
\(|(T_{1}[g])(x)|\leq M_{1}(p)\|g\|_{L^{p}(B)}\),
-
(2)
\(|(T_{1}[g])(x^{(1)})-(T_{1}[g])(x^{(2)})| \leq M_{2}(p)\|g\|_{L^{p}(B)}|x^{(1)}-x^{(2)}|^{\beta}\), \(x^{(1)},x^{(2)}\in R^{4}\),
-
(3)
\({\partial_{x}(T_{1}[g])(x)=g(x)}\), \(x\in B\), \(\partial_{x}(T_{1}[g])(x)=0\), \(x\in R^{4}\backslash\overline{B}\),
where \({0<\beta=\frac{p-4}{p}<1}\).
Proof
(1) By the Hölder inequality, we have
When \(x\in\overline{B}\), because of \(p>4\), \({\frac {1}{p}+\frac{1}{q}}=1\), then \({1< q<\frac{4}{3}}\). Thus \(\int_{B}\frac{1}{|y-x|^{3q}}|dy|\) is bounded. Hence we suppose
When \(x\in R^{4}-\overline{B}\), by Lemma 2.2 and the generalized spherical coordinate, we have
where \(\rho=|y-x|\), \(d_{0}=d(x, B)\).
Therefore, for arbitrary \(x\in R^{4}\), we can obtain
where \({M_{1}(p)=\max\{J_{1}J_{2}, J_{1}J_{4}\}}\).
(2) For arbitrary \(x^{(1)}, x^{(2)}\in R^{4}\), \(x^{(1)}\neq x^{(2)}\), by the Hile lemma [9] and the Hölder inequality, we can obtain
We suppose \(\alpha'=kq\), \(\beta'=(4-k)q\) (\(k=1,2,3\)). From \({1\leq q<\frac{4}{3}}\), we have
Hence, by Lemma 2.3, we have
So we have
where \(M_{2}(p)=J_{7}\), \({0<\beta=\frac{p-4}{p}<1}\).
(3) For arbitrary \(\varphi\in C_{0}^{\infty}(B,{\mathit{Cl}}_{0,3})\), by Definition 2.1, Lemma 2.1, and the Fubini theorem, we have
Hence, in the sense of generalized derivatives, \(\partial _{x}(T_{1}[g])(x)=g(x)\), \(x\in B\). It is easy to see \(\partial _{x}(T_{1}[g])(x)=0\), \(x\in R^{4}\backslash B\). □
Theorem 3.2
Let B be as stated above. If \(g\in L^{p,4}(R^{4},{\mathit{Cl}}_{0,3})\), \(4< p<+\infty\), then we have the following results:
-
(1)
\(|(T_{2}[g])(x)|\leq M_{3}(p)\|g^{(4)}\| _{L^{p}(B)}\), \(x\in R^{4}\),
-
(2)
\(|(T_{2}[g])(x^{(1)})-(T_{2}[g])(x^{(2)})| \leq M_{4}(p)\|g^{(4)}\|_{L^{p}(B)}|x^{(1)}-x^{(2)}|^{\beta}\), \(x^{(1)},x^{(2)}\in R^{4}\),
-
(3)
\({\partial_{x}(T_{2}[g])(x)=0}\), \(x\in B\), \({\partial_{x}(T_{2}[g])(x)=g(x)}\), \(x\in R^{4}\backslash\overline{B}\),
where \({0<\beta=\frac{p-4}{p}<1}\).
Proof
(1) By the Hölder inequality, we have
where \({\frac{1}{p}+\frac{1}{q}=1}\).
Next we will discuss \(O(x)\) in two cases.
(i) When \({|x|\geq\frac{1}{2}}\), since
we have
Let \(\alpha'=q\), \(\beta'=3q\), by \({1< q<\frac{4}{3}}\). we have
Thus, by Lemma 2.3, we have
(ii) When \({|x|<\frac{1}{2}}\), by \(|y|\leq1\), we have \({|1-yx|\geq\frac{1}{2}}\). Thus
Therefore, by (3.2)-(3.4), we have
where \(M_{3}(p)=\max\{J_{8}J_{10}^{\frac{1}{q}},J_{8}J_{13}^{\frac {1}{q}}\}\).
(2) By the Hile lemma [9], we have
Again, because of
by the Hölder inequality, we have
where
In the following, we discuss \(\widetilde{O}_{k}(x^{(1)},x^{(2)})\) in four cases.
(i) When \(|x^{(1)}|\leq\frac{1}{2}\), \(|x^{(2)}|\leq\frac {1}{2}\), we have \(|1-yx^{(1)}|\geq\frac {1}{2}\), \(|1-yx^{(2)}|\geq\frac{1}{2}\) and \(|x^{(1)}-x^{(2)}|\leq1\). Hence
From \(|x^{(1)}|-|x^{(2)}|\leq1\), \(0\leq\beta=1-\frac {4}{p}<1\), we have \(|x^{(1)}-x^{(2)}|\leq|x^{(1)}-x^{(2)}|^{\beta}\). Therefore, by (3.5), we have
(ii) When \(|x^{(1)}|\geq\frac{1}{2}\), \(|x^{(2)}|\leq\frac {1}{2}\), we have \(|1-yx^{(2)}|\geq\frac{1}{2}\), \(\frac {1}{|x^{(1)}|}\leq2\), \(\frac{|x^{(2)}|}{|x^{(1)}|}\leq1\). Thus
where
Again, since
we have
Again from \({1< q<\frac{4}{3}}\), we have \({kq<4}\) (\(k=1,2,3\)). Thus \(\int_{B}\vert \frac{\bar {x}^{(1)}}{|x^{(1)}|^{2}}-y\vert ^{-kq}|dy|\) is bounded. Hence, we obtain
Therefore, by (3.5), we have
(iii) When \(|x^{(1)}|\leq\frac{1}{2}\), \(|x^{(2)}|\geq \frac{1}{2}\), we have \(|1-yx^{(1)}|\geq\frac {1}{2}\), \(\frac{1}{|x^{(2)}|}\leq2\), \(\frac{|x^{(1)}|}{|x^{(2)}|}\leq 1\). Similar to (ii), we have
(V) When \(|x^{(1)}|\geq\frac{1}{2}\), \(|x^{(2)}|\geq\frac {1}{2}\), we have \(\frac{1}{|x^{(1)}|}\leq2\), \(\frac {1}{|x^{(2)}|}\leq2\).
Since
and
we have
Suppose \(\alpha'=kq\), \(\beta'=(4-k)q\), then \(0<\alpha'<3q<4\), \(0<\beta '<3q<4\), \(\alpha'+\beta'=4q\geq4\). Thus, by Lemma 2.3, we have
Therefore, by (3.5), we have
where \({0<\beta=1+\frac{4(1-q)}{q}=\frac{p-4}{p}<1}\).
Therefore, by (3.6)-(3.9), we obtain
where \(M_{4}(p)=\max\{J_{16},J_{23},J_{24},J_{30}\}\).
(3) This case is similar to Theorem 3.1, and it is easy to prove. □
Remark 3.1
Let B be as stated above. If \(g\in L^{p,4}(R^{4},{\mathit{Cl}}_{0,3})\), \(4< p<+\infty\), then we have the following results:
-
(1)
\(|(T[g])(x)|\leq M_{5}(p)\|g\|_{p,4}\), \(x\in R^{4}\),
-
(2)
\(|(T[g])(x^{(1)})-(T[g])(x^{(2)})|\leq M_{6}(p)\|g\| _{p,4}|x^{(1)}-x^{(2)}|^{\beta}\), \(x^{(1)},x^{(2)}\in R^{4}\),
-
(3)
\({\partial_{x}(T[g])(x)=g(x)}\), \(x\in R^{4}\backslash \partial B\),
where \(0<\beta<1\).
4 Integral representation of solution to inhomogeneous partial differential system
In this section, we will discuss the inhomogeneous partial differential system of first order equations as follows:
where \(w_{j}(x)\), \(c_{j}(x)\) (\(j=0,1,2,\ldots,7\)) are real-value functions.
Problem P
Let \(B\subset R^{4}\) be as stated above. The Riemann boundary value problem for system (4.1) is to find a solution \(w(x)\) of (4.1) that satisfies the boundary condition
where \(w^{\pm}(\tau)=\lim_{x\in B^{\pm}, x\rightarrow\tau}w(x)\), \(B^{+}=B\), \(B^{-}=R^{4}\backslash\overline{B}\), G is a Clifford constant, \(G^{-}\) exists, and \(f\in H_{\partial B}^{\nu}\) (\(0<\nu<1\)).
In fact,
Let
By (4.3) and (4.4), the inhomogeneous partial differential system (4.1) can be transformed to the following equation:
Therefore Problem P as stated above can be transformed to Problem Q.
Problem Q
Let \(B\subset R^{4}\) be as stated above. The Riemann boundary value problem for system (4.1) is to find a solution \(w(x)\) of (4.5) that satisfies the boundary condition
where \(w^{\pm}(\tau)=\lim_{x\in B^{\pm}, x\rightarrow\tau}w(x)\), \(B^{+}=B\), \(B^{-}=R^{4}\backslash\overline{B}\), G is a Clifford constant, \(G^{-}\) exists, and \(f\in H_{\partial B}^{\nu}\) (\(0<\nu<1\)).
Theorem 4.1
Let B be as stated above. Find a Clifford-valued function \(u(x)\) satisfying the system \(\partial_{x}u=0\) (\(x\in R^{4}\backslash\partial B\)) and vanishing at infinity with the boundary condition
where \(u^{\pm}(\tau)=\lim_{x\in B^{\pm}, x\rightarrow\tau}u(x)\), G is a Clifford constant, \(G^{-}\) exists, and \(f\in H_{\partial B}^{\lambda }\) (\(0<\lambda<1\)). Then the solution can be expressed as
Proof
Define
Then it is obvious \(\partial_{x}\varphi(x)=0\), and the Riemann boundary condition (4.6) is equivalent to
Suppose \(\psi(x)=\frac{1}{2\pi^{2}}\int_{\partial\Omega_{x}}\frac{\bar{y} -\bar{x}}{|y-x|^{4}}\,d\sigma_{y}f(y)\), then \(\partial_{x}\psi(x)=0\). And by the Plemelj formula [14], we have
Hence \(\varphi^{+}(\tau)-\psi^{+}(\tau)=\varphi^{-}(\tau)-\psi^{-}(\tau)\) (\(\tau \in\partial B\)). Thus by the Liouville theorem and the extension theorem [17], we obtain \(\varphi(x)=\psi(x)\). So, the solution can be expressed as
 □
Theorem 4.2
Let B be as stated above, \(g\in L^{p,4}(R^{4},{\mathit{Cl}}_{0,3})\), \(4< p<+\infty \). Find a Clifford-valued function \(w(x)\) satisfying the system \(\partial _{x}w=g(x)\) (\(x\in R^{4}\backslash\partial B\)) and vanishing at infinity with the boundary condition
where \(w^{\pm}(\tau)=\lim_{x\in B^{\pm}, x\rightarrow\tau}w(x)\), G is a Clifford constant, \(G^{-}\) exists, and \(f\in H_{\partial B}^{\lambda }\) (\(0<\lambda<1\)). Then the solution has the form
in which \(\partial_{x}\Psi(x)=0\) and
where \(\tilde{f}=f+(T[g])(G-1)\), \((T[g])(x)\) is the same as (3.1).
Proof
By Remark 3.1, we know
The boundary condition (4.7) is equivalent to
Again from Remark 3.1, we know that \((T[g])(x)\) has Hölder continuity in \(R^{4}\). Thus \((T[g])^{+}=(T[g])^{-}=T[g]\). So (4.8) is equivalent to
Suppose \(\tilde{f}=f+T[g](G-1)\), then (4.9) has the following form:
Again from Theorem 4.1, the solutions which satisfy the system \(\partial_{x}\Psi(x)=0\) and boundary condition (4.10) can be represented in the form
Remark 4.1
From Theorem 4.2, the solution of Problem P can be expressed as
in which \(\partial_{x}\Psi(x)=0\) and
where \(\tilde{f}=f+T[g](G-1)\). □
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Acknowledgements
This work was supported by the National Science Foundation of China (No. 11401162, No. 11571089, No. 11401159, No. 11301136) and the Natural Science Foundation of Hebei Province (No. A2015205012, No. A2016205218, No. A2014205069, No. A2014208158) and Hebei Normal University Dr. Fund (No. L2015B03).
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Qiao, Y., Wang, L. & Yang, G. A kind of boundary value problem for inhomogeneous partial differential system. J Inequal Appl 2016, 180 (2016). https://doi.org/10.1186/s13660-016-1121-1
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DOI: https://doi.org/10.1186/s13660-016-1121-1