Let Ω be an open bounded nonempty subset of \(\mathbb{R}^{3}\) with a Lyapunov boundary ∂Ω, \(u(\mathbf{x})=\sum_{A}e_{A}u_{A}(\mathbf{x})\), where \(u_{A}(\mathbf{x})\) are real functions. \(u(\mathbf{x})\) is called a Hölder continuous functions on Ω̅ if the following condition is satisfied:
$$\begin{aligned} \bigl\| u(\mathbf{x}_{1})-u(\mathbf{x}_{2}) \bigr\| = \biggl[\sum _{A}\bigl\| u_{A}(\mathbf{x}_{1})-u_{A}( \mathbf{x}_{2})\bigr\| \biggr]^{\frac{1}{2}}\leq C\|\mathbf{x}_{1}- \mathbf{x}_{2}\|^{\alpha}, \end{aligned}$$
where for any \(\mathbf{x}_{1}, \mathbf{x}_{2}\in\overline{\Omega}\), \(\mathbf{x}_{1}\neq\mathbf{x}_{2}\), \(0<\alpha\leq1\), C is a positive constant independent of \(\mathbf{x}_{1}\), \(\mathbf{x}_{2}\).
Denote by \(H^{\alpha}(\partial\Omega,Cl(V_{3,3}))\) the set of Hölder continuous functions with values in \(Cl(V_{3,3})\) on ∂Ω (the Hölder exponent is α, \(0<\alpha<1\)). Define the norm of u in \(H^{\alpha}(\partial\Omega, Cl(V_{3,3}))\) as
$$\begin{aligned} \|u\|_{(\alpha,\partial\Omega)}=\|u\|_{\infty}+\|u\|_{\alpha}, \end{aligned}$$
(3.1)
where \(\|u\|_{\infty}:=\sup_{\mathbf{x}\in\partial\Omega}\| u(\mathbf{x})\|\), \(\|u\|_{\alpha}:=\mathop{\sup_{{\mathbf{x}_{1},\mathbf {x}_{2}\in\partial\Omega}}}\limits_{\hphantom{aaaa}\mathbf{x}_{1}\neq\mathbf{x}_{2}} \frac{\|u(\mathbf{x}_{1})-u(\mathbf{x}_{2})\|}{\|\mathbf{x}_{1}-\mathbf {x}_{2}\|^{\alpha}}\).
Lemma 3.1
[22]
The Hölder space
\(H^{\alpha}(\partial\Omega,Cl(V_{3,3}))\)
is a Banach space with norm (3.1).
Lemma 3.2
Let
\(f, g\in C^{1}(\Omega, Cl(V_{3,3}))\cap C(\overline{\Omega}, Cl(V_{3,3}))\). Then
$$\begin{aligned} \int _{\partial\Omega}f\,d\sigma_{\mathbf{y}}g= \int _{\Omega }[f]L_{\kappa}g\,dV+ \int _{\Omega}fL_{-\kappa}[g]\,dV = \int _{\Omega}[f]L_{-\kappa}g\,dV+ \int _{\Omega}fL_{\kappa }[g]\,dV. \end{aligned}$$
Proof
From Stokes’ theorem in Clifford analysis in [26], the results can be directly proved. □
Theorem 3.3
If
\(u\in C^{2}(\Omega, Cl(V_{3,3}))\cap C^{1}(\overline{\Omega}, Cl(V_{3,3}))\)
where Ω is an open bounded nonempty subset of
\(\mathbb{R}^{3}\)
with a Lyapunov boundary
∂Ω, then
$$\begin{aligned} & \int _{\partial\Omega}K_{\ast1}(\mathbf{x},\mathbf{y},\kappa ) \,d \sigma_{\mathbf{y}}u(\mathbf{y}) +\frac{1}{4\pi} \int _{\partial\Omega}\frac{e^{-\kappa\|\mathbf {y}-\mathbf{x}\|}}{\|\mathbf{y}-\mathbf{x}\|}\,d\sigma_{\mathbf{y}}L_{\kappa}[u]( \mathbf{y}) \\ &\quad{}-\frac{1}{4\pi} \int _{\Omega} \frac{e^{-\kappa\|\mathbf{y}-\mathbf{x}\|}}{\|\mathbf{y}-\mathbf{x}\| }H[u](\mathbf{y})\,dV= \left \{ \textstyle\begin{array}{@{}l@{\quad}l} u(\mathbf{x}), & \mathbf{x}\in\Omega,\\ 0, & \mathbf{x}\in\mathbb{R}^{3}\setminus\overline{\Omega}, \end{array}\displaystyle \right . \end{aligned}$$
(3.2)
where
\(K_{\ast1}(\mathbf{x},\mathbf{y},\kappa)\)
is as in (2.8).
Proof
Let \(\mathbf{x}\in\mathbb{R}^{3}\setminus\overline{\Omega}\). Using Lemma 3.2, we get
$$\begin{aligned} &\frac{1}{4\pi} \int _{\Omega}\frac{e^{-\kappa\|\mathbf{y}-\mathbf {x}\|}}{\|\mathbf{y}-\mathbf{x}\|}H[u](\mathbf{y})\,dV \\ &\quad=\frac{1}{4\pi} \int _{\partial\Omega}\frac{e^{-\kappa\|\mathbf {y}-\mathbf{x}\|}}{\|\mathbf{y}-\mathbf{x}\|}\,d\sigma_{\mathbf{y}} L_{\kappa}[u](\mathbf{y}) -\frac{1}{4\pi} \int _{\Omega} \biggl[\frac{e^{-\kappa\|\mathbf{y}-\mathbf {x}\|}}{\|\mathbf{y}-\mathbf{x}\|} \biggr]L_{\kappa} L_{\kappa}[u](\mathbf{y})\,dV \\ &\quad=\frac{1}{4\pi} \int _{\partial\Omega}\frac{e^{-\kappa\|\mathbf {y}-\mathbf{x}\|}}{\|\mathbf{y}-\mathbf{x}\|}\,d\sigma_{\mathbf{y}} L_{\kappa}[u](\mathbf{y})+ \int _{\Omega}K_{\ast1}(\mathbf {x},\mathbf{y}, \kappa)L_{\kappa}[u](\mathbf{y})\,dV \\ &\quad=\frac{1}{4\pi} \int _{\partial\Omega}\frac{e^{-\kappa\|\mathbf {y}-\mathbf{x}\|}}{\|\mathbf{y}-\mathbf{x}\|}\,d\sigma_{\mathbf{y}} L_{\kappa}[u](\mathbf{y})+ \int _{\partial\Omega}K_{\ast1}(\mathbf {x},\mathbf{y},\kappa) \,d \sigma_{\mathbf{y}}u(\mathbf{y}). \end{aligned}$$
Then the left-hand side of (3.2) apparently equals zero.
Now, let \(\mathbf{x}\in\Omega\) and take \(r>0\) such that \(B(\mathbf{x}, r)\subset\Omega\). Invoking the previous case, we may then write
$$\begin{aligned} & \int _{\partial(\Omega\setminus B(\mathbf{x}, r))}K_{\ast 1}(\mathbf{x},\mathbf{y},\kappa)\,d \sigma_{\mathbf{y}}u(\mathbf{y}) +\frac{1}{4\pi} \int _{\partial(\Omega\setminus B(\mathbf{x}, r))} \frac{e^{-\kappa\|\mathbf{y}-\mathbf{x}\|}}{\|\mathbf{y}-\mathbf{y}\| }\,d\sigma_{\mathbf{y}}L_{\kappa}[u]( \mathbf{y}) \\ &\quad{}-\frac{1}{4\pi} \int _{\Omega\setminus B(\mathbf{x}, r)}\frac {e^{-\kappa\|\mathbf{y}-\mathbf{x}\|}}{\|\mathbf{y}-\mathbf{y}\|} H[u](\mathbf{y})\,dV=0. \end{aligned}$$
(3.3)
Here we take the limits for \(r\rightarrow0\). As regards the weak singularity of \(\frac{e^{-\kappa\|\mathbf{y}-\mathbf{x}\|}}{\|\mathbf {y}-\mathbf{x}\|}\), the third term of (3.3) yields
$$\begin{aligned} \lim_{r\rightarrow0} \int _{\Omega\setminus B(\mathbf{x}, r)}\frac{e^{-\kappa\|\mathbf{y}-\mathbf{x}\|}}{\|\mathbf{y}-\mathbf{x}\|} H[u](\mathbf{y})\,dV= \int _{\Omega}\frac{e^{-\kappa\|\mathbf {y}-\mathbf{x}\|}}{\|\mathbf{y}-\mathbf{x}\|}H[u](\mathbf{y})\,dV. \end{aligned}$$
(3.4)
Furthermore we write
$$\begin{aligned} & \int _{\partial(\Omega\setminus B(\mathbf{x}, r))}K_{\ast 1}(\mathbf{x},\mathbf{y},\kappa)\,d \sigma_{\mathbf{y}}u(\mathbf{y}) +\frac{1}{4\pi} \int _{\partial(\Omega\setminus B(\mathbf{x}, r))} \frac{e^{-\kappa\|\mathbf{y}-\mathbf{x}\|}}{\|\mathbf{y}-\mathbf{x}\| }\,d\sigma_{\mathbf{y}}L_{\kappa}[u]( \mathbf{y}) \\ &\quad= \int _{\partial\Omega}K_{\ast1}(\mathbf{x},\mathbf{y},\kappa )\,d \sigma_{\mathbf{y}} u(\mathbf{y})+ \frac{1}{4\pi} \int _{\partial\Omega} \frac{e^{-\kappa\|\mathbf{y}-\mathbf{x}\|}}{ \|\mathbf{y}-\mathbf{x}\| }\,d\sigma_{\mathbf{y}}L_{\kappa}[u]( \mathbf{y}) \\ &\qquad{}- \int _{\partial B(\mathbf{x}, r)} K_{\ast1}(\mathbf{x},\mathbf {y},\kappa)\,d \sigma_{\mathbf{y}}u(\mathbf{y}) -\frac{1}{4\pi} \int _{\partial B(\mathbf{x}, r)} \frac{e^{-\kappa\|\mathbf{y}-\mathbf{x}\|}}{ \|\mathbf{y}-\mathbf{x}\| }\,d\sigma_{\mathbf{y}}L_{\kappa}[u]( \mathbf{y}). \end{aligned}$$
(3.5)
We denote
$$\begin{aligned} \Theta(\mathbf{x})\triangleq \int _{\partial B(\mathbf{x}, r)} K_{\ast1}(\mathbf{x},\mathbf {y},\kappa)\,d \sigma_{\mathbf{y}}u(\mathbf{y}) +\frac{1}{4\pi} \int _{\partial B(\mathbf{x}, r)} \frac{e^{-\kappa\|\mathbf{y}-\mathbf{x}\|}}{\|\mathbf{y}-\mathbf{x}\| }\,d\sigma_{\mathbf{y}}L_{\kappa}[u]( \mathbf{y}). \end{aligned}$$
(3.6)
It follows from the Stokes formula that
$$\begin{aligned} \Theta(\mathbf{x})={}&\frac{3e^{-\kappa r}}{4\pi r^{3}} \int _{B(\mathbf{x}, r)}u(\mathbf{y})\,dV +\frac{3\kappa e^{-\kappa r}}{4\pi r^{2}} \int _{B(\mathbf{x}, r)}u(\mathbf{y})\,dV \\ &{}+\frac{e^{-\kappa r}}{4\pi r^{3}} \int _{B(\mathbf{x}, r)}(\mathbf {y}-\mathbf{x})D[u](\mathbf{y})\,dV + \frac{\kappa e^{-\kappa r}}{4\pi r^{2}} \int _{B(\mathbf{x}, r)}(\mathbf{y}-\mathbf{x})D[u](\mathbf{y})\,dV \\ &{}+\frac{e^{-\kappa r}}{4\pi r} \int _{B(\mathbf{x}, r)}\Delta [u](\mathbf{y})\,dV. \end{aligned}$$
(3.7)
Applying the Lebesgue differentiation theorem, we have
$$\begin{aligned} \lim_{r\rightarrow0} \Theta(\mathbf{x})=u(\mathbf{x}). \end{aligned}$$
(3.8)
Combining (3.3) with (3.4)-(3.8), we get the desired result. □
Theorem 3.4
If
\(u\in C^{2}(\Omega, Cl(V_{3,3}))\cap C^{1}(\overline{\Omega}, Cl(V_{3,3}))\)
where Ω is an open bounded nonempty subset of
\(\mathbb{R}^{3}\)
with a Lyapunov boundary
∂Ω, then
$$\begin{aligned} & \int _{\partial\Omega}K_{1}(\mathbf{x},\mathbf{y},\kappa)\,d \sigma _{\mathbf{y}}u(\mathbf{y}) +\frac{1}{4\pi} \int _{\partial\Omega}\frac{e^{-\kappa\|\mathbf {y}-\mathbf{x}\|}}{\|\mathbf{y}-\mathbf{x}\|}\,d\sigma_{\mathbf{y}}L_{-\kappa}[u]( \mathbf{y}) \\ &\quad{}-\frac{1}{4\pi} \int _{\Omega} \frac{e^{-\kappa\|\mathbf{y}-\mathbf{x}\|}}{\|\mathbf{y}-\mathbf{x}\| }H[u](\mathbf{y})\,dV= \left \{ \textstyle\begin{array}{@{}l@{\quad}l} u(\mathbf{x}), & \mathbf{x}\in\Omega,\\ 0, & \mathbf{x}\in\mathbb{R}^{3}\setminus\overline{\Omega}, \end{array}\displaystyle \right . \end{aligned}$$
(3.9)
where
\(K_{1}(\mathbf{x},\mathbf{y},\kappa)\)
is as in (2.7).
Proof
The result can be similarly proved to Theorem 3.3. □
Corollary 3.5
If
\(u\in C^{2}(\Omega, Cl(V_{3,3}))\cap C^{1}(\overline{\Omega}, Cl(V_{3,3}))\)
where Ω is an open bounded nonempty subset of
\(\mathbb{R}^{3}\)
with a Lyapunov boundary
∂Ω and
\(H[u]=L_{\kappa}L_{-\kappa}[u]=0\)
in Ω, then
$$ \int _{\partial\Omega}K_{1}(\mathbf{x},\mathbf{y},\kappa)\,d \sigma _{y}u(\mathbf{y}) +\frac{1}{4\pi} \int _{\partial\Omega}\frac{e^{-\kappa\|\mathbf {y}-\mathbf{x}\|}}{\|\mathbf{y}-\mathbf{x}\|}\,d\sigma_{\mathbf{y}}L_{-\kappa}[u]( \mathbf{y}) =\left \{ \textstyle\begin{array}{@{}l@{\quad}l} u(\mathbf{x}), & \mathbf{x}\in\Omega,\\ 0, & \mathbf{x}\in\mathbb{R}^{3}\setminus\overline{\Omega}, \end{array}\displaystyle \right . $$
(3.10)
where
\(K_{1}(\mathbf{x},\mathbf{y},\kappa)\)
is as in (2.7).
Corollary 3.6
If
\(u\in C^{2}(\Omega, Cl(V_{3,3}))\cap C^{1}(\overline{\Omega}, Cl(V_{3,3}))\)
where Ω is an open bounded nonempty subset of
\(\mathbb{R}^{3}\)
with a Lyapunov boundary
∂Ω and
\(H[u]=L_{-\kappa}L_{\kappa}[u]=0\)
in Ω, then
$$ \int _{\partial\Omega}K_{\ast1}(\mathbf{x},\mathbf{y},\kappa ) \,d \sigma_{y}u(\mathbf{y}) +\frac{1}{4\pi} \int _{\partial\Omega}\frac{e^{-\kappa\|\mathbf {y}-\mathbf{x}\|}}{\|\mathbf{y}-\mathbf{x}\|}\,d\sigma_{\mathbf{y}}L_{\kappa}[u]( \mathbf{y}) =\left \{ \textstyle\begin{array}{@{}l@{\quad}l} u(\mathbf{x}), & \mathbf{x}\in\Omega,\\ 0, & \mathbf{x}\in\mathbb{R}^{3}\setminus\overline{\Omega}, \end{array}\displaystyle \right . $$
(3.11)
where
\(K_{\ast1}(\mathbf{x},\mathbf{y},\kappa)\)
is as in (2.8).
Corollary 3.7
Let
\(f(\mathbf{x})\in C_{c}^{2}(\Omega, Cl(V_{3,3}))\). The solution of the following Dirichlet boundary value problem:
$$\begin{aligned} \left \{ \textstyle\begin{array}{@{}l@{\quad}l} Hu=f, & \textit{in }\Omega,\\ L_{\kappa}[u]=0, & \textit{on }\partial\Omega,\\ u=0, & \textit{on }\partial\Omega, \end{array}\displaystyle \right . \end{aligned}$$
(3.12)
is
$$\begin{aligned} u(\mathbf{x})=-\frac{1}{4\pi} \int_{\Omega}\frac{e^{-\kappa\|\mathbf {y}-\mathbf{x}\|}}{\|\mathbf{y}-\mathbf{x}\|}f(\mathbf{y})\,dV. \end{aligned}$$
(3.13)
Proof
By Theorem 3.3, the solution of (3.12) is formulated as
$$\begin{aligned} u(\mathbf{x}) =& \int_{\partial\Omega}K_{\ast1}(\mathbf{x}, \mathbf{y}, \kappa) \,d \sigma_{\mathbf{y}}u(\mathbf{y}) +\frac{1}{4\pi} \int _{\partial\Omega}\frac{e^{-\kappa\|\mathbf {y}-\mathbf{x}\|}}{\|\mathbf{y}-\mathbf{x}\|}\,d\sigma_{\mathbf{y}}L_{\kappa}[u]( \mathbf{y}) \\ &{}-\frac{1}{4\pi} \int _{\Omega} \frac{e^{-\kappa\|\mathbf{y}-\mathbf{x}\|}}{\|\mathbf{y}-\mathbf{x}\| }H[u](\mathbf{y})\,dV, \end{aligned}$$
(3.14)
since \(L_{\kappa}[u]=0\) and \(u=0\) on ∂Ω, the result follows. □
Using Theorem 3.4, we also have the following result which can be similarly proved to Corollary 3.7.
Corollary 3.8
Let
\(f(\mathbf{x})\in C_{c}^{2}(\Omega, Cl(V_{3,3}))\). The solution of the Dirichlet boundary value problem
$$\begin{aligned} \left \{ \textstyle\begin{array}{@{}l@{\quad}l} Hu=f, & \textit{in }\Omega,\\ L_{-\kappa}[u]=0, & \textit{on }\partial\Omega,\\ u=0, & \textit{on }\partial\Omega, \end{array}\displaystyle \right . \end{aligned}$$
(3.15)
is
$$\begin{aligned} u(\mathbf{x})=-\frac{1}{4\pi} \int_{\Omega}\frac{e^{-\kappa\|\mathbf {y}-\mathbf{x}\|}}{\|\mathbf{y}-\mathbf{x}\|}f(\mathbf{y})\,dV. \end{aligned}$$
(3.16)
Next, we introduce the following generalized Teodorescu operators \(\mathbb{T}_{\pm\kappa}\), the generalized Cauchy integral operators \(\mathbb{F}_{\pm\kappa}\), and the generalized Cauchy singular integral operators \(\mathbb{S}_{\pm\kappa}\):
$$\begin{aligned}& \mathbb{T}_{\kappa}[u](\mathbf{x})\triangleq- \int_{\Omega}K_{\ast 1}(\mathbf{x},\mathbf{y},\kappa)u( \mathbf{y})\,dV, \quad \mathbf {x}\in\Omega, \end{aligned}$$
(3.17)
$$\begin{aligned}& \mathbb{T}_{-\kappa}[u](\mathbf{x})\triangleq- \int_{\Omega}K_{1}(\mathbf {x},\mathbf{y},\kappa)u( \mathbf{y})\,dV, \quad \mathbf{x}\in\Omega, \end{aligned}$$
(3.18)
$$\begin{aligned}& \mathbb{F}_{\kappa}[u](\mathbf{x})\triangleq \int_{\partial\Omega}K_{\ast 1}(\mathbf{x},\mathbf{y},\kappa)\,d \sigma_{y}u(\mathbf{y}), \quad \mathbf{x}\in\mathbb{R}^{3} \setminus\partial\Omega, \end{aligned}$$
(3.19)
$$\begin{aligned}& \mathbb{F}_{-\kappa}[u](\mathbf{x})\triangleq \int_{\partial\Omega }K_{1}(\mathbf{x},\mathbf{y},\kappa)\,d \sigma_{y}u(\mathbf{y}), \quad\mathbf{x}\in\mathbb{R}^{3} \setminus\partial\Omega, \end{aligned}$$
(3.20)
$$\begin{aligned}& \mathbb{S}_{\kappa}[u](\mathbf{x})\triangleq2 \int_{\partial\Omega }K_{\ast1}(\mathbf{x},\mathbf{y},\kappa)\,d \sigma_{y}u(\mathbf{y}), \quad \mathbf{x}\in\partial \Omega, \end{aligned}$$
(3.21)
$$\begin{aligned}& \mathbb{S}_{-\kappa}[u](\mathbf{x})\triangleq2 \int_{\partial\Omega }K_{1}(\mathbf{x},\mathbf{y},\kappa)\,d \sigma_{y}u(\mathbf{y}), \quad \mathbf{x}\in\partial \Omega, \end{aligned}$$
(3.22)
where \(\kappa\geq0\), \(u\in H^{\alpha}(\partial\Omega,Cl(V_{3,3}))\).
Lemma 3.9
[22]
Let Ω be an open nonempty bounded subset of
\(\mathbb{R}^{3}\)
with a Lyapunov boundary
∂Ω, \(u\in H^{\alpha}(\partial\Omega,Cl(V_{3,3}))\), \(0<\alpha\leq1\). Then
$$\begin{aligned}& \mathop{\lim_{\mathbf{x}\rightarrow\mathbf{x}_{0}\in\partial\Omega}}_{\mathbf{x}\in\Omega}\mathbb{F}_{\kappa}[u]( \mathbf{x})= \frac{u(\mathbf {x}_{0})}{2}+\frac{1}{2} \mathbb{S}_{\kappa}[u]( \mathbf{x}_{0}), \end{aligned}$$
(3.23)
$$\begin{aligned}& \mathop{\lim_{\mathbf{x}\rightarrow\mathbf{x}_{0}\in\partial\Omega}}_{ \mathbf{x}\in\mathbb{R}^{3}\setminus\overline{\Omega}}\mathbb{F}_{\kappa }[u]( \mathbf{x})=- \frac{u(\mathbf{x}_{0})}{2}+\frac{1}{2} \mathbb{S}_{\kappa}[u]( \mathbf{x}_{0}), \end{aligned}$$
(3.24)
$$\begin{aligned}& \mathop{\lim_{\mathbf{x}\rightarrow\mathbf{x}_{0}\in\partial\Omega}}_{ \mathbf{x}\in\Omega}\mathbb{F}_{-\kappa}[u]( \mathbf{x})= \frac{u(\mathbf {x}_{0})}{2}+ \frac{1}{2}\mathbb{S}_{-\kappa}[u]( \mathbf{x}_{0}), \end{aligned}$$
(3.25)
$$\begin{aligned}& \mathop{\lim_{\mathbf{x}\rightarrow\mathbf{x}_{0}\in\partial\Omega}}_{ \mathbf{x}\in\mathbb{R}^{3}\setminus\overline{\Omega}}\mathbb {F}_{-\kappa}[u]( \mathbf{x})=- \frac{u(\mathbf{x}_{0})}{2}+\frac{1}{2} \mathbb{S}_{-\kappa}[u]( \mathbf{x}_{0}). \end{aligned}$$
(3.26)
Theorem 3.10
Let Ω be an open bounded non-empty subset of
\(\mathbb{R}^{3}\)
with a Lyapunov boundary
∂Ω, \(u\in C^{1}(\Omega, Cl(V_{3,3}))\cap C(\overline{\Omega},Cl(V_{3,3}))\). Then for
\(\mathbf{x}\in\Omega\),
$$\begin{aligned} L_{\kappa}\mathbb{T}_{\kappa}[u](\mathbf{x})=u( \mathbf{x}). \end{aligned}$$
(3.27)
Proof
Step 1. Because \(u(\mathbf{x})\) has its compact support \(\operatorname{supp}[u]\Subset\Omega\), we have
$$\begin{aligned} \mathbb{T}_{\kappa}[u](\mathbf{x}) =&- \int_{\Omega}K_{\ast1}(\mathbf {x},\mathbf{y},\kappa)u( \mathbf{y})\,dV \\ =&- \int_{\mathbb{R}^{3}}K_{\ast1}(\mathbf{x},\mathbf{y},\kappa)u( \mathbf {y})\,dV \\ =&- \int_{\mathbb{R}^{3}}K_{\ast1}(\mathbf{x},\mathbf{y}+\mathbf {x}, \kappa)u(\mathbf{y}+\mathbf{x})\,dV. \end{aligned}$$
In view of \(u(\mathbf{x})\) having a compact support, the operator \(L_{\kappa}\) acting on \(\mathbb{T}_{\kappa}[u](\mathbf{x})\) may be interchanged with integration. Thus we get
$$\begin{aligned} L_{\kappa}\mathbb{T}_{\kappa}[u]( \mathbf{x}) =&-\lim _{r\rightarrow 0} \int_{\mathbb{R}^{3}\setminus B(0,r)} \Biggl[\sum_{i=1}^{3}e_{i} \frac{\partial}{\partial x_{i}} \bigl[K_{\ast 1}(\mathbf{x},\mathbf{y}+\mathbf{x}, \kappa)u(\mathbf{y}+\mathbf{x}) \bigr] \\ &{}+\kappa K_{\ast1}(\mathbf{x},\mathbf{y}+\mathbf{x},\kappa)u( \mathbf{y}+ \mathbf {x}) \Biggr]\,dV \\ =&-\lim_{r\rightarrow0} \int_{\mathbb{R}^{3}\setminus B(0,r)} \Biggl[\sum_{i=1}^{3}e_{i}K_{\ast1}( \mathbf{x},\mathbf{y}+\mathbf {x},\kappa)\frac{\partial}{\partial x_{i}}u(\mathbf{y}+ \mathbf{x}) \\ &{}+\kappa K_{\ast1}(\mathbf{x},\mathbf{y}+\mathbf{x},\kappa)u( \mathbf{y}+ \mathbf {x}) \Biggr]\,dV \\ =&-\lim_{r\rightarrow0} \int_{\mathbb{R}^{3}\setminus B(0,r)} \Biggl[\sum_{i=1}^{3}e_{i}K_{\ast1}( \mathbf{x},\mathbf{y}+\mathbf {x},\kappa)\frac{\partial}{\partial y_{i}}u(\mathbf{y}+ \mathbf{x}) \\ &{} +\kappa K_{\ast1}(\mathbf{x},\mathbf{y}+\mathbf{x},\kappa)u( \mathbf{y}+ \mathbf {x}) \Biggr]\,dV \\ =&-\lim_{r\rightarrow0} \int_{\mathbb{R}^{3}\setminus B(0,r)} \Biggl[\sum_{i=1}^{3}e_{i} \frac{\partial}{\partial y_{i}} \bigl[K_{\ast 1}(\mathbf{x},\mathbf{y}+\mathbf{x}, \kappa)u( \mathbf{y}+\mathbf{x}) \bigr] \Biggr]\,dV. \end{aligned}$$
Using the Stokes formula, we conclude that
$$\begin{aligned} L_{\kappa}\mathbb{T}_{\kappa}[u]( \mathbf{x}) =&\lim _{r\rightarrow 0} \int_{\|\mathbf{y}\|=r}\,d\sigma_{\mathbf{y}} K_{\ast1}(\mathbf{x}, \mathbf{y}+\mathbf{x},\kappa)u(\mathbf{y}+\mathbf {x}) \\ =&\lim_{r\rightarrow0} \int_{\|\mathbf{y}\|=r}\,d\sigma_{\mathbf{y}} K_{\ast1}(\mathbf{x}, \mathbf{y}+\mathbf{x},\kappa) \bigl[u(\mathbf{y}+\mathbf {x})-u(\mathbf{x}) \bigr] \\ &{}+\lim_{r\rightarrow0} \int_{\|\mathbf{y}\|=r}\,d\sigma_{\mathbf{y}} K_{\ast1}(\mathbf{x}, \mathbf{y}+\mathbf{x},\kappa)u(\mathbf{x}) \\ =&\lim_{r\rightarrow0}\frac{1}{4\pi} \int_{\|\mathbf{y}\| =r}\,d\sigma_{\mathbf{y}} \biggl(\frac{\mathbf{y}}{\|\mathbf{y}\|^{3}} + \frac{\kappa\mathbf{y}}{\|\mathbf{y}\|^{2}}-\frac{\kappa}{\|\mathbf {y}\|} \biggr)e^{-\kappa\|\mathbf{y}\|}u(\mathbf{x}) \\ =&\lim_{r\rightarrow0} \biggl(\frac{3e^{-\kappa r}}{4\pi r^{3}} \int_{\|\mathbf{y}\|\leq r}\,dV+\frac{3\kappa e^{-\kappa r}}{4\pi r^{2}} \int_{\|\mathbf{y}\|\leq r}\,dV \biggr)u(\mathbf{x}) \\ =&u(\mathbf{x}). \end{aligned}$$
(3.28)
Thus we have proved that (3.27) follows for any \(u(\mathbf{x})\in C_{c}^{1}(\Omega, Cl(V_{3,3}))\).
Step 2. We prove that (3.27) holds for any \(u(\mathbf {x})\in C^{1}(\Omega, Cl(V_{3,3}))\). We take a neighborhood V of x such that \(\mathbf{x}\in V\Subset\Omega\), a real-valued function \(\Psi\in C^{\infty}(\Omega)\) such that \(\Psi|_{V}=1\) and \(\operatorname{supp}\Psi\Subset\Omega\). Then
$$\begin{aligned} u(\mathbf{x})=u\Psi+u(1-\Psi):=u_{1}(\mathbf{x})+u_{2}( \mathbf{x}). \end{aligned}$$
It is obvious that \(u_{1}(\mathbf{x})\in C_{c}^{1}(\Omega, Cl(V_{3,3}))\), \(u_{2}(\mathbf{x})\in C^{1}(\Omega, Cl(V_{3,3}))\) and \(u_{1}|_{V}=u\), \(u_{2}|_{V}=0\). Following step 1, we obtain
$$\begin{aligned} L_{\kappa}\mathbb{T}_{\kappa}[u_{1}]( \mathbf{x})=u_{1}(\mathbf {x})=u(\mathbf{x}),\quad \mathbf{x}\in V. \end{aligned}$$
(3.29)
Since \(u_{2}(\mathbf{x})\) equals zero in V, we get
$$\begin{aligned} L_{\kappa}\mathbb{T}_{\kappa}[u_{2}]( \mathbf{x})&=L_{\kappa} \biggl[- \int _{\Omega}K_{\ast1}(\mathbf{x}, \mathbf{y}, \kappa)u_{2}(\mathbf {y})\,dV \biggr] \\ &=L_{\kappa} \biggl[- \int_{\Omega\setminus V}K_{\ast1}(\mathbf{x}, \mathbf {y}, \kappa)u_{2}(\mathbf{y})\,dV \biggr] \\ &=0. \end{aligned}$$
(3.30)
It follows from (3.29) and (3.30) that
$$\begin{aligned} L_{\kappa}\mathbb{T}_{\kappa}[u](\mathbf{x})=u(\mathbf{x}). \end{aligned}$$
(3.31)
Because x is taken arbitrarily in Ω, the result follows. □
Corresponding to Theorem 3.10, we have the following theorem.
Theorem 3.11
Let Ω be an open bounded non-empty subset of
\(\mathbb{R}^{3}\)
with a Lyapunov boundary
∂Ω, \(u\in C^{1}(\Omega, Cl(v_{3,3}))\cap C(\overline{\Omega},Cl(V_{3,3}))\). Then for
\(\mathbf{x}\in\Omega\),
$$\begin{aligned} L_{-\kappa}\mathbb{T}_{-\kappa}[u](\mathbf{x})=u( \mathbf{x}). \end{aligned}$$
(3.32)
In the following, we need to consider Hölder’s boundedness of the singular integral operators \(\mathbb{S}_{\pm\kappa}\). It is necessary to solve the following boundary value problems in Clifford analysis.
Theorem 3.12
Let Ω be an open nonempty bounded subset of
\(\mathbb{R}^{3}\)
with a Lyapunov boundary
∂Ω. Then the generalized Cauchy integral operator
\(\mathbb{S}_{\kappa}\): \(H^{\alpha}(\partial\Omega,Cl(V_{3,3}))\mapsto H^{\alpha}(\partial\Omega,Cl(V_{3,3}))\)
defined by (3.21) is bounded, i.e.
$$\begin{aligned} \bigl\| \mathbb{S}_{\kappa}[u]\bigr\| _{(\alpha, \partial\Omega)} \leq C\|u\| _{(\alpha, \partial\Omega)}, \end{aligned}$$
(3.33)
where
\(C=\max \{\frac{C_{4}}{2\pi}\eta^{2-\alpha}(|\partial\Omega |+\frac{1}{2-\alpha})+C_{6}, \frac{C_{5}(|\partial\Omega|+\eta^{2}+\eta)}{2\pi\eta}+\frac{2\kappa (C_{2}\eta^{2}+C_{3}|\partial\Omega|)}{\eta} \}\)
and
\(|\partial\Omega|\)
denotes the surface area of Ω.
Proof
For \(\mathbf{x}\in\partial\Omega\), we have
$$\begin{aligned} \bigl\| \mathbb{S}_{\kappa}[u]( \mathbf{x})\bigr\| ={}&\biggl\| \frac{2}{4\pi} \int_{\partial \Omega} \biggl[\frac{\mathbf{y}-\mathbf{x}}{\|\mathbf{y}-\mathbf{x}\|^{3}} +\frac{\kappa(\mathbf{y}-\mathbf{x})}{\|\mathbf{y}-\mathbf{x}\| ^{2}}- \frac{\kappa}{\|\mathbf{y}-\mathbf{x}\|} \biggr]e^{-\kappa\|\mathbf {y}-\mathbf{x}\|}\,d\sigma_{\mathbf{y}}u(\mathbf{y})\biggr\| \\ \leq{}&\biggl\| \frac{1}{2\pi} \int_{\partial\Omega}\frac{\kappa(\mathbf {y}-\mathbf{x})}{\|\mathbf{y}-\mathbf{x}\|^{2}} e^{-\kappa\|\mathbf{y}-\mathbf{x}\|}\,d \sigma_{\mathbf{y}}u(\mathbf{y})\biggr\| \\ &{}+\biggl\| \frac{1}{2\pi} \int_{\partial\Omega}\frac{\kappa}{\|\mathbf {y}-\mathbf{x}\|} e^{-\kappa\|\mathbf{y}-\mathbf{x}\|}\,d \sigma_{\mathbf{y}} u(\mathbf{y})\biggr\| \\ &{}+\biggl\| \frac{1}{2\pi} \int_{\partial\Omega}\frac{\mathbf{y}-\mathbf{x}}{\| \mathbf{y}-\mathbf{x}\|^{3}} e^{-\kappa\|\mathbf{y}-\mathbf{x}\|}\,d \sigma_{\mathbf{y}}u(\mathbf{y})\biggr\| \\ :={}&J_{1}+J_{2}+J_{3}. \end{aligned}$$
(3.34)
Since ∂Ω is Lyapunov boundary, the normal vector n is continuous on ∂Ω. Therefore, we can choose \(0<\eta\leq1\) such that for the scalar product
$$\begin{aligned} \bigl(\mathbf{n}(\mathbf{x});\mathbf{n}(\mathbf{y}) \bigr)\geq \frac{1}{2} \end{aligned}$$
(3.35)
for all \(\mathbf{x}, \mathbf{y}\in\partial\Omega\) with \(\|\mathbf {y}-\mathbf{x}\|\leq\eta\). It is enough to consider the case of \(\|\mathbf{y}-\mathbf{x}\|\) being sufficiently small such that the set \(\partial L \triangleq\{\mathbf{y}\in\partial\Omega: \|\mathbf {y}-\mathbf{x}\|\leq\eta\}\) is connected for each \(\mathbf{x}\in\partial\Omega\). Then the condition (3.35) implies that ∂L can be bijective into the tangent plane to ∂Ω at the point x. Using polar coordinates \((r, \omega)\) in the tangent plane with origin in x, for any \(u\in H^{\alpha}(\partial\Omega, Cl(V_{3,3}))\), we arrive at
$$\begin{aligned} \biggl\| \int_{\partial L}\frac{\kappa(\mathbf{y}-\mathbf{x})}{\|\mathbf {y}-\mathbf{x}\|^{2}}e^{-\kappa\|\mathbf{y}-\mathbf{x}\|}\,d\sigma _{\mathbf{y}}u(\mathbf{y})\biggr\| &\leq\kappa C_{1}\|u\|_{\infty} \int_{\partial L}\frac{1}{\|\mathbf {y}-\mathbf{x}\|}\,dS \leq2\pi\kappa C_{2}\|u\|_{\infty} \int_{0}^{\eta}\,dr \\ &=2\pi\kappa C_{2}\eta\|u\|_{\infty}, \end{aligned}$$
(3.36)
where \(C_{1}\), \(C_{2}\) denote nonnegative constants which are independent of u. Here we use the facts that \(\|\mathbf{x}-\mathbf{y}\|\geq r\), that the surface element
$$\begin{aligned} dS=\frac{r\,dr\,d\omega}{(\mathbf{n}(\mathbf{x}),\mathbf{x}(\mathbf{y}))} \end{aligned}$$
(3.37)
can be estimated with the aid of (3.35) by \(dS\leq2r\,dr\, d\omega\), and that the projection ∂L into the tangent plane is contained in the interior of the sphere of radius η and center x. Furthermore,
$$\begin{aligned} \biggl\| \int_{\partial\Omega\setminus\partial L}\frac{\kappa(\mathbf {y}-\mathbf{x})}{\|\mathbf{y}-\mathbf{x}\|^{2}}e^{-\kappa\|\mathbf {y}-\mathbf{x}\|}\,d \sigma_{\mathbf{y}}u(\mathbf{y})\biggr\| &\leq2\pi\kappa C_{3}\|u \|_{\infty} \int_{\partial\Omega\setminus \partial L}\eta^{-1}\,dS \\ &\leq2\pi\kappa C_{3}\|u\|_{\infty}\eta^{-1}|\partial \Omega|, \end{aligned}$$
(3.38)
where \(|\partial\Omega|\) is the surface area of Ω. Inequalities (3.36) with (3.38) imply
$$\begin{aligned} J_{1}\leq\frac{\kappa C_{2}\eta^{2}+\kappa C_{3}|\partial\Omega|}{\eta }\|u\|_{\infty}. \end{aligned}$$
(3.39)
Using a similar method to the proof of \(J_{1}\), we obtain
$$\begin{aligned} J_{2}\leq\frac{\kappa C_{2}\eta^{2}+\kappa C_{3}|\partial\Omega|}{\eta }\|u\|_{\infty}. \end{aligned}$$
(3.40)
Now we estimate \(J_{3}\). Combining \(u\in H^{\alpha}(\partial\Omega, Cl(V_{3,3}))\) with
$$\frac{1}{4\pi} \int_{\partial\Omega}\frac{\mathbf{y}-\mathbf{x}}{\| \mathbf{y}-\mathbf{x}\|^{3}}\,d\sigma_{\mathbf{y}}= \frac{1}{2}, \quad\mbox{for }\mathbf{x}\in\partial\Omega, $$
we have
$$\begin{aligned} J_{3} \leq&\biggl\| \frac{1}{2\pi} \int_{\partial\Omega}\frac{\mathbf{y}-\mathbf {x}}{\|\mathbf{y}-\mathbf{x}\|^{3}}\,d\sigma_{\mathbf{y}} \bigl[e^{-\kappa\|\mathbf{y}-\mathbf{x}\|} \bigl(u(\mathbf{y})- u(\mathbf{x}) \bigr) \bigr]\biggr\| \\ &{}+\biggl\| \frac{1}{2\pi} \int_{\partial\Omega}\frac{\mathbf{y}-\mathbf{x}}{\| \mathbf{y}-\mathbf{x}\|^{3}}\,d\sigma_{\mathbf{y}} \bigl[u( \mathbf{x})e^{-\kappa\|\mathbf{y}-\mathbf{x}\|}-u(\mathbf{x}) \bigr]\biggr\| +\bigl\| u(\mathbf{x})\bigr\| \\ \leq& C_{4}\|u\|_{\alpha}\frac{1}{2\pi} \int_{\partial\Omega}\frac{1}{\| \mathbf{y}-\mathbf{x}\|^{2-\alpha}}\,dS +C_{5} \frac{1}{2\pi}\|u\|_{\infty} \int_{\partial\Omega}\frac{1}{\| \mathbf{y}-\mathbf{x}\|}\,dS +\|u\|_{\infty} \\ \leq&\frac{C_{4}}{2\pi} \biggl( \int_{\partial\Omega\setminus\partial L}\frac {1}{\|\mathbf{y}-\mathbf{x}\|^{2-\alpha}}\,dS+ \int_{\partial L}\frac{1}{\| \mathbf{y}-\mathbf{x}\|^{2-\alpha}}\,dS \biggr)\|u\|_{\alpha} \\ &{}+\frac{C_{5}}{2\pi} \biggl( \int_{\partial\Omega\setminus\partial L}\frac{1}{\| \mathbf{y}-\mathbf{x}\|}\,dS+ \int_{\partial L}\frac{1}{\|\mathbf {y}-\mathbf{x}\|}\,dS \biggr)\|u\|_{\infty} + \|u \|_{\infty} \\ \leq&\frac{C_{4}}{2\pi} \biggl(\eta^{2-\alpha}|\partial\Omega|+ \frac{\eta ^{2-\alpha}}{2-\alpha} \biggr)\|u\|_{\alpha} +\frac{C_{5}}{2\pi} \bigl( \eta^{-1}|\partial\Omega|+\eta+1 \bigr)\|u\|_{\infty}. \end{aligned}$$
(3.41)
Combining (3.34), (3.39), (3.40), and (3.41), we get
$$\begin{aligned} \bigl\| \mathbb{S}_{\kappa}[u]\bigr\| _{\infty} \leq& \frac{C_{4}}{2\pi}\eta ^{2-\alpha} \biggl(|\partial\Omega|+\frac{1}{2-\alpha} \biggr)\|u\|_{\alpha} \\ &{}+ \biggl[\frac{C_{5}(|\partial\Omega|+\eta^{2}+\eta)}{2\pi\eta}+\frac{2\kappa (C_{2}\eta^{2}+C_{3}|\partial\Omega|)}{\eta} \biggr]\|u\|_{\infty}, \end{aligned}$$
(3.42)
where \(C_{4}\), \(C_{5}\), \(C_{6}\) denote nonnegative constants which are independent of u.
On the other hand, for \(\mathbf{x}_{1}\), \(\mathbf{x}_{2}\in\partial \Omega\), it is enough to consider the case of \(\|\mathbf{x}_{1}-\mathbf{x}_{2}\|\) being sufficiently small. It is obvious that
$$\begin{aligned} e^{-\kappa\|\mathbf{y}-\mathbf{x}_{i}\|}u(\mathbf{y}) \in H^{\alpha} \bigl(\partial\Omega\times \partial\Omega, Cl(V_{3,3}) \bigr),\quad i=1,2, \end{aligned}$$
and
$$\begin{aligned} \|\mathbf{y}-\mathbf{x}\|e^{-\kappa\|\mathbf{y}-\mathbf{x}_{i}\| }u(\mathbf{y})\in H^{\alpha} \bigl( \partial\Omega\times\partial\Omega, Cl(V_{3,3}) \bigr), \quad i=1,2. \end{aligned}$$
Applying some properties of the Hilbert transform in Clifford analysis (see [8, 9, 16, 30]) and the weak singularity of \(\frac{\mathbf{y}-\mathbf{x}}{\|\mathbf{y}-\mathbf{x}\|^{2}}\) and \(\frac {1}{\|\mathbf{y}-\mathbf{x}\|}\), we conclude
$$\begin{aligned} \bigl\| \mathbb{S}_{\kappa}[u](\mathbf{x}_{1})- \mathbb{S}_{\kappa}[u](\mathbf {x}_{2})\bigr\| \leq C_{6}\|u \|_{\alpha}\|\mathbf{x}_{1}-\mathbf{x}_{2}\| ^{\alpha}, \end{aligned}$$
(3.43)
where \(C_{6}\) is a nonnegative constant independent of \(\mathbf{x}_{1}\) and \(\mathbf{x}_{2}\). It follows from (3.42) and (3.43) that
$$\begin{aligned} \bigl\| \mathbb{S}_{\kappa}[u]\bigr\| _{(\alpha, \partial\Omega)}\leq C\|u\| _{(\alpha, \partial\Omega)}, \end{aligned}$$
where \(C=\max\{\frac{C_{4}}{2\pi}\eta^{2-\alpha}(|\partial\Omega|+\frac {1}{2-\alpha})+C_{6}, \frac{C_{5}(|\partial\Omega|+\eta^{2}+\eta)}{2\pi\eta}+\frac{2\kappa (C_{2}\eta^{2}+C_{3}|\partial\Omega|)}{\eta}\}\). The proof is complete. □
Remark 3.13
By the same technique we obtain the following result that the generalized Cauchy integral operator \(\mathbb{S}_{-\kappa}\): \(H^{\alpha }(\partial\Omega,Cl(V_{3,3}))\mapsto H^{\alpha}(\partial\Omega,Cl(V_{3,3}))\) defined by (3.22) is bounded.
Remark 3.14
We assume \(u\in H^{\alpha}(\partial\Omega, Cl(V_{3,3}))\). All integrals are understood in the Riemann integral sense in Lemma 3.9 and Theorem 3.12. Now, let \(L^{p}(\partial\Omega, Cl(V_{3,3}))\), \(1\leq p<\infty\) be the space of all Clifford algebra valued functions, whose pth power is Lebesgue integrable in ∂Ω. If \(u\in L^{p}(\partial\Omega, Cl(V_{3,3}))\) then one has to understand \(\mathbb{F}_{\pm\kappa}\) as a Lebesgue integral, and the necessary changes can be easily made. For instance, the limits exist almost everywhere on ∂Ω with respect to the surface Lebesgue measure in Lemma 3.9. Using classical Calderón-Zygmund theory, an \(L^{p}\) formulation of Theorem 3.12 holds.
In the framework of Clifford algebra \(Cl(V_{3,3})\), we come back to the modified Helmholtz equation \((\Delta-\kappa^{2})[u](\mathbf{x})=0\), \(\mathbf{x}\in\Omega\). By Theorem 3.3, Theorem 3.4, Theorem 3.10, and Theorem 3.11, we have the following theorem.
Theorem 3.15
Suppose that Ω is an open nonempty bounded subset of
\(\mathbb {R}^{3}\)
with a Lyapunov boundary
∂Ω, \(f, g\in C^{1}(\Omega, Cl(V_{3,3}))\cap C(\overline{\Omega}, Cl(V_{3,3}))\), \(L_{-\kappa}[f]=0\)
and
\(L_{\kappa}[g]=0\)
in Ω. Then the function
\(u(\mathbf{x})\)
is determined by
$$\begin{aligned} u(\mathbf{x})=\mathbb{T}_{\kappa}[f](\mathbf{x})+g( \mathbf{x}) \end{aligned}$$
(3.44)
or
$$\begin{aligned} u(\mathbf{x})=\mathbb{T}_{-\kappa}[g](\mathbf{x})+f( \mathbf{x}). \end{aligned}$$
(3.45)
Conversely, suppose
\(u(\mathbf{x})\in C^{1}(\overline{\Omega}, Cl(V_{3,3}))\)
and
\(u(\mathbf{x})\)
is a solution of the modified Helmholtz equation. Then
u
may be represented by (3.44) or (3.45), where
\(L_{-\kappa}[f]=0\)
and
\(L_{\kappa}[g]=0\)
in Ω.