Basic Properties of Solutions of Boundary Value Problem for the Dirac-Wave Operator

Abstract

We study local and global properties of solutions of the boundary value problem for the Dirac wave operator. This includes explicit formulas, domain of dependence, range of influence, finite propagation speed and energy estimates. We also introduce the space of Dirac wave homogeneous polynomials and the corresponding Dirac-wave conjugate functions.

1 Introduction

As it is well know from a basic course in PDE, solutions of the boundary value problem (BVP) for the wave operator or d’Alembertian $\Box {:}= \partial ^2_t -\Delta =0$ have as main local properties explicit formulas, domain of dependence, range of influence, finite propagation speed and as global properties energy estimates among others. On the other hand, we also recall that the fundamental solution of the wave operator is not $C^{\infty }$ and this has important implication for the regularity of their solutions. In particular, a singularity at a point propagates along the characteristic cone and focuses at its vertex.

The Dirac-wave operator is a first order linear operator with coefficients in a clifford algebra which factor the wave operator. We will present analogues properties as the ones mentioned above for the wave operator for solutions of the BVP for the Dirac-wave operator. This is a first step to have a kind of complex analysis for the wave operator. We also will present the corresponding wave conjugate functions and its Hilbert transform and the space of homogeneous polynomials which satisfy the Dirac-wave operator.

Previous works from David Eelbode and Frank Sommen compute the fundamental solution for the Dirac-wave operator on hyperbolic spaces [1,2,3] see also Rosen [4] Chapter 9. As we said we will be interested in local and global properties of solutions. We point out that the behavior of solutions of the hyperbolic operator $\Box$ are rather different of solutions of the elliptic operator $\Delta$ and the same will applies to solutions of the Dirac operator and the Dirac-wave operator.

As an introduction and for future reference let us recall the BVP for the wave operator and its solution. We will follow [5,6,7]. Recall that the BVP for the wave operator is to find $u \in C^{2}$ $(0, \infty )\times {\mathbb {R}}^n$ such that

$$\begin{aligned} \begin{array}{l} \Box u(t,x)=0 \\ u(0,x)=\phi (x) \ \ \ \ \partial _t u(0,x)=\psi (x) \ \ x \in {\mathbb {R}}^n, \end{array} \end{aligned}$$

(1.1)

given the initial value $\phi (x)$ and the initial velocity $\psi (x)$.

We state the solution to the BVP in dimension three first and comment about the other cases below.

Theorem 1.1

(Kirchhoff formula) Let $k\ge 2$ be an integer, $\phi (x) \in C^{k+1}({\mathbb {R}}^3)\$ and $\ \psi (x) \in C^k({\mathbb {R}}^3).$ Suppose u is defined by

$$\begin{aligned} u(t,x)= \frac{1}{4\pi t} \int _{\partial B(x,t)}\psi (y)dS(y)+\partial _t\left( \ \frac{1}{4\pi t} \int _{\partial B(x,t)}\phi (y)dS(y) \ \right) \end{aligned}$$

(1.2)

then $u \in C^k((0, \infty )\times {\mathbb {R}}^3)$ and satisfy the wave operator:

$$\begin{aligned} \partial ^2_t u(t,x)-\Delta u(t,x)=0 \ \ \ \text{ in } \ \ \ (0, \infty )\times {\mathbb {R}}^3. \end{aligned}$$

Moreover, for any $x_0 \in {\mathbb {R}}^3$

$$\begin{aligned} \lim _{(t,\, x)\rightarrow (0, \, x_0)} u(t,x)=\phi (x_0) \ \ \ \lim _{(t,\, x)\rightarrow (0, \, x_0)} \partial _t u(t,x)=\psi (x_0). \end{aligned}$$

(1.3)

Note that the solution to the BVP u is $C^k$-continuous up to the boundary. Given $k\ge 2$ and any $\varphi (x) \in C^{k+1}({\mathbb {R}}^3)$ it is defined its Riemann transform on $(0, \infty )\times {\mathbb {R}}^3$ as

$$\begin{aligned} ( {\mathcal {R}} \varphi )(t,x)=\frac{1}{4\pi t} \int _{\partial B(x,t)}\varphi (y)dS(y). \end{aligned}$$

(1.4)

We will need that following identity which is easy to see

$$\begin{aligned} \frac{\partial }{\partial x_j}{\mathcal {R}} \varphi (t,x)={\mathcal {R}}\left( \frac{\partial \varphi }{\partial x_j}\right) (t,x) \ \ \ j=1,2,3. \end{aligned}$$

(1.5)

It follows that Kirchhoff formula could be written as

$$\begin{aligned} u(t,x)= ( {\mathcal {R}} \psi )(t,x)+ \partial _t ( {\mathcal {R}} \phi )(t,x). \end{aligned}$$

(1.6)

Also note that ${\mathcal {R}} \psi$ solves the BVP with zero initial values and with initial velocity $\psi$ and $\partial _t ( {\mathcal {R}} \phi )$ solves the BVP with initial values $\phi$ and with zero initial velocity.

Now a few comments on the other dimensions. When $n > 3$ is odd the explicit formula for the solution of the BVP need $\phi (x) \in C^{\frac{n-1}{2}+k}({\mathbb {R}}^n)\$ and $\psi (x) \in C^{\frac{n-3}{2}+k}({\mathbb {R}}^n)$ with $k\ge 2$ and the corresponding Riemann transform of $\varphi$ is given by

$$\begin{aligned} {} ({\mathcal {R}}_{odd} \varphi )(t,x)=\frac{1}{c_n} \left[ \left( \frac{1}{t} \frac{\partial }{\partial t}\right) ^{\frac{n-3}{2}}\left( \frac{1}{\omega _n \, t} \int _{\partial B(x,t)}\varphi (y)dS(y)\right) \right] \end{aligned}$$

(1.7)

where $c_n=1\cdot 3 \cdots (n-2)$ and $\omega _n$ is the surface area of the unit sphere in ${\mathbb {R}}^n$. Then the solution of the BVP is given in an analogous way as in (1.6). The solution for the case when n is even is derived from the solution for $n+1$ by the method of decent. The corresponding transform is given by

$$\begin{aligned} {} ({\mathcal {R}}_{even} \varphi )(t,x)=\frac{2}{c_{n+1}} \left[ \left( \frac{1}{t} \frac{\partial }{\partial t}\right) ^{\frac{n-2}{2}}\left( \frac{n}{\omega _{n+1} \, t} \int _{B(x,t)}\frac{\varphi (y)}{\sqrt{t^2-\Vert y-x\Vert ^2}} dy\right) \right] . \nonumber \\ \end{aligned}$$

(1.8)

Again the solution of the BVP in the even case is given in an analogous way as in (1.6). This explicit formulas (1.7), (1.8) demonstrate the necessity of making more and more smoothness assumption on the data to ensure the existence of a $C^2$-solution. The key point here is we can work directly with $C^{\infty }$-functions as data. For the distributional case we recommended [8].

The layout of this note is as follows. In Sect. 2 we introduce the Clifford algebra and the Dirac-wave operator and some of their properties. We also introduce the Teodorescu transform that we will use later. In Sect. 3 we study the BVP for the Dirac-wave operator. We give several equivalences of their solutions. Here we see the domain of dependences and the range of influences of the wave-monogenic functions. As an application we find a basic for the space of Dirac-wave homogeneous polynomials. In Sect. 4 we study the energy estimates for the Dirac-wave operator. In particular, we see here uniqueness for the Dirac wave equation in a bounded set and finite propagation speed. In the last Sect. 5, we see how to extend a set of wave functions to a monogenic-wave function and from this we define the corresponding Dirac-wave conjugate function. It turns out that the Teodorescu transform corresponds to the wave Hilbert transform. We also decompose any solution of the wave BVP with values in the Clifford algebra as a sum of a Dirac-wave BVP solution and an anti Dirac-wave BVP solution.

2 Clifford Algebras and Dirac Operators

Here we collect basic properties of the Clifford algebra with will be working with. For more background material and further general references on Clifford algebras and related matters, the interested reader is referred to the monographs [9,10,11]. See also [4].

We start with the Minkowski space ${\mathbb {R}}^{1,n}$, this is ${\mathbb {R}}^{n+1}=\big \{(t,x_1,x_2,\dots ,x_n)\big \}$ with the quadratic form

$$\begin{aligned} Q( \ (t,x) \ )=t^2-\Vert x\Vert ^2, \end{aligned}$$

(2.1)

and with the standard orthonormal basis $\{\epsilon ,e_1, e_2, \dots , e_n\}$. The Clifford algebra ${{\mathcal {C}}}\,\ell (1,n)$ with signature (1, n) is the minimum enlargement of ${\mathbb {R}}^{1,n}$ to a unitary algebra. This implies that the following relations holds

$$\begin{aligned} \epsilon \,\epsilon =1, \ \ \ \ \ e_j\, e_j=-1, \ \ \ \ \epsilon \, e_j= -e_j\, \epsilon ,\ \ \ \ j=1, \dots , n, \ \ \ \ e_i\, e_j= -e_j \, e_i \ \ \ i\ne j. \nonumber \\ \end{aligned}$$

(2.2)

Any element $u\in {{\mathcal {C}}}\,\ell (1,n)$ can be uniquely represented in the form

$$\begin{aligned} u=\sum _{I}u_I\,e_I=\sum _{\ell =0}^{n+1}{\sum _{|I|=\ell }}'u_I\,e_I,\qquad u_I\in {{\mathbb {C}}}. \end{aligned}$$

(2.3)

Here $e_I$ stands for the product $e_{i_1}\, e_{i_2}\,\cdots \, e_{i_\ell }$ if $I=(i_1,i_2,\dots ,i_\ell )$ and $e_0{:}=\epsilon$, $e_{\emptyset }{:}=1$ is the multiplicative unit. Also, $\sum '$ indicates that the sum is performed only over strictly increasing multi-indices, i.e., l-tuples $I=(i_1,i_2,\dots ,i_\ell )$ with $0\le i_1<i_2<\dots <i_\ell \le n$.

We will use properties of the standard Clifford algebra ${\mathcal {C}}\,\ell (0,n)$. In particular, any $u\in {{\mathcal {C}}}\,\ell (1,n)$ in the form

$$\begin{aligned} u=u^1+ \epsilon u^2 \end{aligned}$$

(2.4)

with $u^1$ and $u^2 \in {{\mathcal {C}}}\,\ell (0,n)$.

We endow ${{\mathcal {C}}}\,\ell (1,n)$ with the semi-inner product structure

$$\begin{aligned} \langle u,v\rangle {:}=\sum _I a_Ic_I-\sum _I b_Id_I,\,\ u=a+\epsilon b,\,\,v=c+\epsilon d \end{aligned}$$

(2.5)

with $a,b,c,d \in {{\mathcal {C}}}\,\ell (0,n)$. If we now define a complex conjugation on ${{\mathcal {C}}}\,\ell (1,n)$ by setting $u^c{:}={\sum _I}u^c_Ie_I$ for each $u={\sum _I}u_Ie_I$, where $z^{\,c}$ denotes the usual complex conjugation of $z\in {{\mathbb {C}}}$ then this semi-inner structure induces the semi-norm

$$\begin{aligned} \Vert u\Vert _{M}{:}=\langle u,u^c\rangle = \sum _I|a_I|^2-\sum _I |b_I|^2,\,\ u=a+\epsilon b. \end{aligned}$$

(2.6)

The Clifford conjugacion on ${{\mathcal {C}}}\,\ell (1,n)$, denoted by ’bar’, is defined as the extension on the conjugacion on ${{\mathcal {C}}}\,\ell (0,n)$ such that $\overline{\epsilon }=-\epsilon \,$ and $\, \overline{\epsilon u}={\overline{u}}\, \overline{\epsilon }$ for any $u\in {{\mathcal {C}}}\,\ell (1,n)$. Thus if $u=u_1+\epsilon u_2$ with $u_1$ and $u_2 \in {{\mathcal {C}}}\,\ell (0,n)$ then ${\overline{u}}= \overline{u_1+\epsilon u_2}= \overline{u_1}- \overline{u_2}\,\epsilon$. More specifically, recall that on ${\mathcal {C}}\,\ell (0,n)$ $\overline{e_I}\, e_I=e_I\,\overline{e_I}=1$ for any multi-index I, $I=(i_1,i_2,\dots ,i_\ell )$ with $1\le i_1<i_2<\dots <i_\ell \le n$ and we have that on ${\mathcal {C}}\,\ell (1,n)$, $\, \overline{\epsilon e_I}\,\epsilon e_I=\epsilon e_I\,\overline{\epsilon e_I}=-1$.

One can also verify without difficulty that

$$\begin{aligned} \overline{{\overline{u}}}=u,\quad \Vert {\overline{u}}\,\Vert _M=\Vert u\Vert _M,\,\,\text { and }\,\,\overline{u\, v}={\overline{v}}\,{\overline{u}}, \qquad \forall \,u,v\in {{\mathcal {C}}}\,\ell (1,n). \end{aligned}$$

(2.7)

Next, for each $\ell \in \{0,1,\ldots ,n+1\}$ consider the projection map $\Pi ^{\ell }$ onto the $\ell$-homogeneous part of u, i.e.,

$$\begin{aligned} \Pi ^{\ell }\,u {:}= {\sum _{|I|=\ell }}'u_I e_I\,\,\,\, \text{ if } \,\,\,\, u=\sum _{\ell =0}^{n+1}{\sum _{|I|=\ell }}'u_I\,e_I\in {\mathcal {C}}\,\ell (1,n), \end{aligned}$$

(2.8)

and denote by $\Lambda ^\ell$ the range of $\Pi ^{\ell }:{\mathcal {C}}\,\ell (1,n) \rightarrow {\mathcal {C}}\,\ell (1,n)$. It follows that

$$\begin{aligned} {\mathcal {C}}\,\ell (1,n)=\Lambda ^0\oplus \Lambda ^1\oplus \dots \oplus \Lambda ^{n+1}. \end{aligned}$$

(2.9)

We will use the subalgebra of even homogeneous element of ${\mathcal {C}}\,\ell (1,n)$ which we denote by ${\mathcal {C}}\,\ell ^{+}(1,n){:}=\Lambda ^0\oplus \Lambda ^2\oplus \dots \oplus \Lambda ^{[{n+1}]}$ (Here [m] denotes the even number small o equal to m).

As an example and for future references we have that any $u\in {\mathcal {C}}\,\ell ^{+}(1,3)$ can be written as

$$\begin{aligned} u{} & {} =\sum u_{I}e_{I}= u_0+ u_{01} \epsilon e_1 +u_{02} \epsilon e_2+ u_{03} \epsilon e_3\nonumber \\{} & {} \quad +u_{23} e_{23} +u_{13} e_{13} +u_{12} e_{12}+u_{0123}\epsilon e_{123}. \end{aligned}$$

(2.10)

Moving on, recall that if we let $\Omega$ be an open set in ${{\mathbb {R}}}^n$. Then the classical (homogeneous) Dirac operator associated with ${{\mathbb {R}}}^n$ is given by

$$\begin{aligned} {\mathbb {D}}{:}=\sum _{j=1}^n e_j\,\partial _j. \end{aligned}$$

(2.11)

This acts on some $u\in C^1(\Omega ,{{\mathcal {C}}}\,\ell (0,n))$ (or in ${\mathcal {C}}\,\ell (1,n)$) according to

$$\begin{aligned} {\mathbb {D}}u{:}=\sum _{j=1}^n e_j\,\partial _j u. \end{aligned}$$

(2.12)

Call u monogenic in $\Omega$ if $\mathbb {D}u=0$ in $\Omega$. The fundamental solution of ${\mathbb {D}}$ is given by

$$\begin{aligned} E(x)=\frac{-1}{\omega _n }\frac{x}{\Vert x\Vert ^n} \ \ \ x \in {\mathbb {R}}^{n}\setminus \{0\}. \end{aligned}$$

(2.13)

We will use, only for functions of compact support, the Teodorescu transform that we recall. For a function $\varphi (x)$ defined in ${{\mathbb {R}}}^n$ with values in ${\mathcal {C}}\,\ell (1,n)$ of compact support the Teodorescu transform of $\varphi (x)$ is defined as

$$\begin{aligned} T(\varphi )(x){:}=- \int _{{\mathbb {R}}^n} E(x-y)\varphi (y)dv(y)\ \ \ x \in {\mathbb {R}}^{n} \end{aligned}$$

(2.14)

The main property of T that we will use [11] Theorem 8.2 is that

$$\begin{aligned} {\mathbb {D}}T(\varphi )(x)= \varphi (x)\ \ \ x \in {\mathbb {R}}^{n}. \end{aligned}$$

(2.15)

Note that the property above is true for ${\mathcal {C}}\,\ell (0,n)$-valued functions but it is easy to see that it extends to ${\mathcal {C}}\,\ell (1,n)$-valued functions.

Now the Dirac-wave operator or Hyperbolic-Dirac operator associated with ${\mathbb {R}}^{1,n}$, is given by

$$\begin{aligned} {\mathbb {D}}_w{:}=\epsilon \partial _t - {\mathbb {D}} \end{aligned}$$

(2.16)

acting on $u\in {{C}}^2((0, T) \times \Omega ,{\mathcal {C}}\,\ell (1,n) \, )$ according to

$$\begin{aligned} {\mathbb {D}}_w u(t,x){:}=\epsilon \partial _t u(t,x) - {\mathbb {D}}u(t,x), \end{aligned}$$

(2.17)

where $T \in (0, \infty ]$.

Call u wave-monogenic in $(0, T) \times \Omega$ if ${\mathbb {D}}_w u=0$ in $(0, T) \times \Omega$. We will always act ${\mathbb {D}}_w$ on the left. There is a parallel theory where ${\mathbb {D}}_w$ acts on the right. This follows from the observation that ${\mathbb {D}}_w u=0$ if and only if ${\overline{u}}{\mathbb {D}}_w =0$ (${\mathbb {D}}_w$ acting on the right).

One of the most fundamental properties of Dirac operators introduced above is that they can be thought of as square-roots of familiar second-order differential operators. More precisely, ${\mathbb {D}}$ and ${\mathbb {D}}_w$ satisfy

$$\begin{aligned} {\mathbb {D}}^2=-\Delta \,\,\,\, \text{ and } \,\,\,{\mathbb {D}}_w^2=\Box , \end{aligned}$$

(2.18)

Another operator of interest will be $\widetilde{{\mathbb {D}}_w}{:}=\epsilon \partial _t +{\mathbb {D}}$, which also satisfy $\widetilde{{\mathbb {D}}_w}^2=\Box$. Moreover as a consecuence of ${\mathbb {D}}\epsilon =-\epsilon {\mathbb {D}}$ it is easy to see that

$$\begin{aligned} {\mathbb {D}}_w u=0 \Longleftrightarrow \widetilde{{\mathbb {D}}_w}(\epsilon u)=0. \end{aligned}$$

(2.19)

If $\widetilde{{\mathbb {D}}_w}(u)=0$ we call u anti wave-monogenic.

We will be interesting in study the space of solutions of the Dirac-wave operator in ${\mathbb {R}}^+ \times {\mathbb {R}}^n$, namely

$$\begin{aligned} {\mathbb {D}}_w ({\mathbb {R}}^+ \times {\mathbb {R}}^n)=\{u\in {C}^2(\,(0, \infty ) \times {\mathbb {R}}^n,{\mathcal {C}}\,\ell (1,n)\, ): {\mathbb {D}}_w u=0 \} \end{aligned}$$

(2.20)

A few remarks that can be verify without difficulty are in order here.

Remark 2.1

The following properties hold

• (i) We requiere wave-monogenic functions to be at least ${C}^2$ in order to satisfy the wave operator.

• (ii) The space ${\mathbb {D}}_w ({\mathbb {R}}^+ \times {\mathbb {R}}^n)$ is a right ${\mathcal {C}}\,\ell (1,n)$ module.

• (iii) If u(t, x) is a scalar-valued solution of the wave operator then ${\mathbb {D}}_w u(t,x)$ is a wave-monogenic function.

• (iv) For any $\phi :{\mathbb {R}}^n \rightarrow {\mathcal {C}}\,\ell (1,n)$, harmonic then $u(t,x)= \phi (x)+t\epsilon {\mathbb {D}}\phi (x)$ is wave-monogenic function. In particular, if $\phi (x)$ is monogenic $u(t,x)= \phi (x)$ is time independent.

• (v) Any solution of ${\mathbb {D}}_w u(t,x)=0$ in ${\mathcal {C}}\,\ell (1,n)$ can be decompose in two independent ${\mathbb {D}}_w$-solutions $u(t,x)=u_1(t,x)+u_2(t,x)$ with $u_1(t,x) \in {\mathcal {C}}\,\ell ^{+}(1,n)$ and $u_2(t,x)\in {\mathcal {C}}\,\ell (1,n){\setminus }{\mathcal {C}}\,\ell ^{+}(1,n)$.

• (vi) Let $u:(0, \infty ) \times {\mathbb {R}}^n\rightarrow {\mathcal {C}}\,\ell (1,n)$ and decompose u as in (2.4), this is $u(t,x)=u^1(t,x)+\epsilon u^2(t,x)$ then ${\mathbb {D}}_w u(t,x)=0$ if and only if $u^1$ and $u^2$ satisfy the system

  $$\begin{aligned} \partial _t u^1(t,x)= & {} -{\mathbb {D}}u^2(t,x) \nonumber \\ \partial _t u^2(t,x)= & {} {\mathbb {D}}u^1(t,x). \end{aligned}$$

  (2.21)

As a key example we present, when n=3, that the whole Maxwell system could be seen as a Dirac-wave function: Let $\vec {E}(t,x),\vec {B}(t,x)$ be the electric field and magnetic field, in a vacuum, the Maxwell system is given by the following four equations:

$$\begin{aligned} \textrm{curl}\,\vec {E}+\frac{\partial \vec {B}}{\partial t}&=0, \nonumber \\ \textrm{curl}\, \vec {B}-\frac{\,\,\partial \vec {E}}{ \partial t}&=-4\pi \vec {J}, \nonumber \\ div \,\vec {E}&=4\pi \rho , \nonumber \\ div\, \vec {B}&=0. \end{aligned}$$

(2.22)

Here $\rho$ is the charge density and $\vec {J}$ the electric current. Units are chosen so that the speed of light is 1. See for example [12] page 165.

We could see the Maxwell system as a nonhomogeneous solution of ${\mathbb {D}}_{w}$ in the following way: If

$$\begin{aligned} u{:}=E_1 \epsilon e_1 +E_2 \epsilon e_2+ E_3 \epsilon e_3+B_1 e_{23} -B_2 e_{13}+B_3 e_{12} \ \ \in \Lambda ^{2} \end{aligned}$$

(2.23)

and

$$\begin{aligned} {\hat{J}}{:}= \rho \epsilon +J_1 e_1+J_2 e_2+J_3 e_3. \end{aligned}$$

(2.24)

A simply computation shows that Maxwell system (2.22) is equivalent to ${\mathbb {D}}_{w}u={\hat{J}}$. In particular, if there is no charge or current then (2.22) is equivalent to ${\mathbb {D}}_{w}u=0$.

3 Boundary Value Problem for the Dirac-Wave Operator

Here we discuss the initial-value problem for the Dirac-wave operator. We will work with data functions of class at least $C^2$ and in fact we could assume $C^{\infty }$. We will be concerned with local properties of solutions of the initial value problem for the Dirac-wave operator.

Given $\phi :{\mathbb {R}}^n \rightarrow {\mathcal {C}}\,\ell (1,n)$ a sufficiently smooth function. Consider the initial value problem for the Dirac-wave equation: Find $u\in {{C}}^2((0, \infty ) \times {\mathbb {R}}^n,{\mathcal {C}}\,\ell (1,n))$ with

$$\begin{aligned} \begin{array}{l} {\mathbb {D}}_w u(t,x)=0 \\ u(0,x)=\phi (x). \end{array} \end{aligned}$$

(3.1)

Recall that we want our solution to be ${C}^2$-continuous up to the boundary. From the remark (2.1)(v) we see that is it enough to take $\phi (x)$ with values in ${\mathcal {C}}\,\ell ^{+}(1,n)$ and u will take values also in ${\mathcal {C}}\,\ell ^{+}(1,n)$.

The case when $n=1$, the d’Alembert formula is simply but nevertheless gives us intuition about what to expect in the case when $n>1$. Consider $\phi (x) = f(x)+\epsilon e_1 g(x)$ and $u(t,x)= u_0(t,x)+\epsilon e_1 u_{01}(t,x)$ to be $C^1$ and $C^2$-function in ${\mathcal {C}}\,\ell ^{+}(1,1)$ respectively. Then it is straightforward to see that the following are equivalent.

• u(t,x) solves (3.1) for $n=1.$

• u satisfy the wave Cauchy-Riemann system

  $$\begin{aligned} \frac{\partial u_0}{\partial t}= \frac{\partial u_{01}}{\partial x} \ \ \ \ \ \frac{\partial u_0}{\partial x}= \frac{\partial u_{01}}{\partial t} \end{aligned}$$

  (3.2)

  and $u(0,x)=f(x)+\epsilon e_1 g(x)$.

• Each component of u satisfy the BVP for $\Box u=0$

  $$\begin{aligned} \begin{array}{l} u_0(0,x)=f(x) \ \ \ \ \ \partial _t u_0(0,x)=g'(x) \\ u_{01}(0,x)=g(x) \ \ \ \ \ \partial _t u_{01}(0,x)=f'(x). \end{array} \end{aligned}$$

  (3.3)

• u(t, x) is given by any of the formulas

  $$\begin{aligned} u(t,x)&=\frac{1}{2}\big (f(x+t)+f(x-t)+g(x+t)-g(x-t)\big )\nonumber \\&\quad +\frac{\epsilon e_1 }{2}\big (g(x+t)+g(x-t)+f(x+t)-f(x-t)\big ) \nonumber \\&=\frac{1}{2}\big (\, \phi (x+t)+\phi (x-t)\big ) +\frac{1}{2}\epsilon e_1 \big (\, \phi (x+t)-\phi (x-t)\big ) \nonumber \\&=\frac{1}{2}(1+\epsilon e_1)\phi (x+t)+\frac{1}{2}(1-\epsilon e_1)\phi (x-t). \end{aligned}$$

  (3.4)

All the standard properties (slightly better) really follows. For example, the domain of dependence of u(t, x) are the endpoints of the interval $[x-t, x+t]$ and the range of influence of an initial value $\phi (x_0)$ is the boundary of the wedge-shaped region $\{ (t,x): x_0-t< x < x_0+t \}$.

For most part of this section we shall entirely restrict our attention to the physically most relevant case, namely $n=3$. All other cases are done in a analogous way using with n is even (1.8) and when $n > 3$ is odd (1.7). Nevertheless we emphasize the necessity of making more and more smoothness assumptions on the initial data to ensure the existence of a ${C}^2$-solution.

First note that the Riemann transform act naturally in Clifford valued functions componentwise. i.e. if $\phi :{\mathbb {R}}^3 \rightarrow {\mathcal {C}}\,\ell (1,3)$, then

$$\begin{aligned} {\mathcal {R}} \phi (t,x) = {\mathcal {R}}\left( \sum _{I}e_I\,\phi _I\right) (t,x)=\sum _{I}e_I\,({\mathcal {R}} \phi _I(t,x)). \end{aligned}$$

(3.5)

In the next Theorem we find the solution to (3.1) and several equivalencies.

Theorem 3.1

Let $\phi :{\mathbb {R}}^3 \rightarrow {\mathcal {C}}\,\ell ^{+}(1,3)$ be at least ${C}^3$ and u(t, x) be a ${C}^2$-function in ${\mathcal {C}}\,\ell ^{+}(1,3)$. Then the following are equivalent

• (i) u(t, x) solves (3.1) for $n=3.$

• (ii) $u(0,x)=\phi (x)$ and u satisfy in $(0, \infty ) \times {\mathbb {R}}^3$ the 8x8 first order differential system

  $$\begin{aligned} \begin{array}{cc} \partial _t u_{0}-\partial _1 u_{01}-\partial _2 u_{02}-\partial _3 u_{03}&{} =0 \\ \partial _t u_{01}-\partial _1 u_{0}-\partial _2 u_{12}-\partial _3 u_{13} &{} =0 \\ \partial _t u_{02}+\partial _1 u_{12}-\partial _2 u_{0}-\partial _3 u_{23} &{} =0 \\ \partial _t u_{03}+\partial _1 u_{13}+\partial _2 u_{23}-\partial _3 u_{0} &{} =0 \\ \partial _t u_{12}+\partial _1 u_{02}-\partial _2 u_{01}-\partial _3 u_{0123} &{} =0 \\ \partial _t u_{13}+\partial _1 u_{03}+\partial _2 u_{0123}-\partial _3 u_{01} &{} =0 \\ \partial _t u_{23}-\partial _1 u_{0123}+\partial _2 u_{03}-\partial _3 u_{02} &{} =0 \\ \partial _t u_{0123}-\partial _1 u_{23}+\partial _2 u_{13}-\partial _3 u_{12} &{} =0. \end{array} \end{aligned}$$

• (iii) For each multi-index I, $u_{I}$ satisfy the BVP    $\Box u_{I}=0$

  $$\begin{aligned} u_{I}(0,x)=\phi _{I}(x) \ \ \ \ \ \partial _t u_{I}(0,x)=\big ( \epsilon {\mathbb {D}} \phi (x)\big )_{I}. \end{aligned}$$

  (3.6)

• (iv) u(t, x) is given by the formula

  $$\begin{aligned} u(t,x)= \partial _t (\, ({\mathcal {R}} \phi )(t,x)\, )+{\mathcal {R}}\big ( \epsilon {\mathbb {D}} \phi \big ) (t,x). \end{aligned}$$

  (3.7)

• (v) If $\phi$ and u are decompose as in (2.4) then

  $$\begin{aligned} u^1(t,x)= & {} \partial _t (\, ( {\mathcal {R}} \phi ^1)(t,x)\, )-{\mathcal {R}}\big ( {\mathbb {D}} \phi ^2 \big ) (t,x) \end{aligned}$$

  (3.8)
  $$\begin{aligned} u^2(t,x)= & {} \partial _t (\, ( {\mathcal {R}} \phi ^2)(t,x)\, )+{\mathcal {R}}\big ( {\mathbb {D}} \phi ^1 \big ) (t,x). \end{aligned}$$

  (3.9)

Proof

[(i)] $\iff$ [(ii)] This is just to write down u as in (2.10) and the equation ${\mathbb {D}}_w u(t,x)=0$ componentwise.

[(i)] $\implies$ [(iii)] We know that $\Box u=0$ taking components give us that for each multi-index I, $\Box u_{I}=0$ and $u_{I}(0,x)=\phi _{I}(x).$ Note that ${\mathbb {D}}_w u=0$ is equivalent to $\partial _t u(t,x) =\epsilon {\mathbb {D}} u(t,x)$ taking the limit as $t \rightarrow 0$ give the condition on the velocity.

[(iii)] $\implies$ [(iv)] From Kirchhoff formula (1.6) we have that for each multi-index I

$$\begin{aligned} u_{I}(t,x)= {\mathcal {R}} ( \epsilon {\mathbb {D}} \phi )_{I}(t,x)+ \partial _t \big (\, {\mathcal {R}}\phi _{I} (t,x)\, \big ). \end{aligned}$$

(3.10)

This yields (3.7).

[(iv)] $\iff$ [(v)] From ${\mathbb {D}}\epsilon =-\epsilon {\mathbb {D}}$ and the decomposition of $\phi$ in (3.7) it follows (3.8). Also from (2.1) (vi) it follows (3.7).

[(iv)] $\implies$ [(i)] Note that $\partial _t ( {\mathcal {R}} \phi )(t,x)$ has initial value $\phi (x)$ and that ${\mathcal {R}}\big (\, \epsilon {\mathbb {D}} \phi \, \big ) (t,x)$ has zero initial value. It follows that u has initial value $\phi (x)$. Next due to (1.5) we have that ${\mathcal {R}} ( \epsilon {\mathbb {D}} \phi )=\epsilon {\mathbb {D}}{\mathcal {R}}\phi$ then u could be writen as

$$\begin{aligned} u(t,x) = \partial _t ( {\mathcal {R}} \phi )(t,x)+\epsilon {\mathbb {D}}{\mathcal {R}}\phi (t,x) =(\partial _t-{\mathbb {D}}\epsilon )({\mathcal {R}} \phi )(t,x)=\big ({\mathbb {D}}_w {\mathcal {R}} \phi (t,x)\big )\epsilon .\nonumber \\ \end{aligned}$$

(3.11)

This yield (3.1). $\square$

Several consequences of this Theorem follows. We collect the first in a form of a remark.

Remark 3.2

The following properties follows for Dirac-wave solutions of (3.1)

• (i) Domain of dependence: u(t, x) depends only on the values of $\phi (y)$ and ${\mathbb {D}}\phi (y)$ for $y \in \partial B(x,t)$. Specifically, for each multi-index I, $u_{I}(t,x)$ depends only on the values of $\phi _{I}(y)$ and $(\epsilon {\mathbb {D}}\phi (y))_{I}$ for $y \in \partial B(x,t)$.

• (ii) Range of influence: $\phi (y)$ influence u(t, x) only on the surface $\Vert x-y\Vert =t$. Specifically, for each multi-index I, $\phi _{I}(y)$ influence the components $u_{I}(t,x)$, $u_{I_1}(t,x),u_{I_2}(t,x),u_{I_3}(t,x)$, only on the surface $\Vert x-y\Vert =t$, where $I_j$ is such that $\epsilon e_j e_{I_j}=\pm e_{I}$ for $j=1,2,3$. By way of example, $\phi _{01}(y)$ influence the components $u_{01}(t,x), u_{0}(t,x), u_{12}(t,x), u_{13}(t,x)$ on the surface $\Vert x-y\Vert =t$.

In the next corollary we will see conditions on which the solution to the BVP is of a particular form. Also we note that if the initial data has only certain components the solution u will have more components in general.

Corollary 3.3

Suppose the initial data $\phi$ for the BVP (3.1) is given as

$$\begin{aligned} \phi (x) = \phi _0(x) +F_1(x) \epsilon e_1 +F_2(x) \epsilon e_2 +F_3(x) \epsilon e_3. \end{aligned}$$

(3.12)

This means all the other components are zero. Assume that $\phi$ is zero at infinity, consider the vector field $F=(F_i)$, if u is the solution to (3.1) then

• If $curl \,F=0$ there exist a scalar valued $\Phi$ wave-solution such that $u(t,x)= (D_w \Phi (t,x)) \epsilon$.

• if $curl \, F$ is not zero u has more that 4 components not zero.

• if $Div F=0$ and $\phi _0\equiv 0$ then u has values in $\Lambda ^{2}$ and they satisfy a Maxwell system.

Proof

The proof is straightforward. $\square$

Here we consider n arbitrary again. We will give an application of the results above.

For any k integer let $P_k({\mathbb {R}}^n, {\mathcal {C}}\,\ell (1,n) )$ denote the space of homogeneous polynomials of degree k with values in ${\mathcal {C}}\,\ell (1,n)$. We will denote by $W_k({\mathbb {R}}\times {\mathbb {R}}^n)$ the space of homogeneous wave-monogenic polynomials of degree k with values in ${\mathcal {C}}\,\ell (1,n)$. We will find a nice basis for $W_k({\mathbb {R}}\times {\mathbb {R}}^n)$.

For each multiindex $\alpha =(k_1,k_2,\dots ,k_n) \in {\mathbb {N}}^n$ with $|\alpha |{:}=k_1+k_2+\dots +k_n = k$ let $p_{\alpha }(x)=x^{\alpha }=x_{1}^{k_1}x_{2}^{k_2}\dots x_{n}^{k_n}$. Then it is well know that a basis of $P_k({\mathbb {R}}^n, {\mathbb {R}} )$ is given by this set of $p_{\alpha }$. Moving on, let $P_{\alpha }(t,x)$ be the solution of the BVP (3.1) with initial data $p_{\alpha }(x)$. We find an expression of $P_{\alpha }$ as follows.

Let

$$\begin{aligned} W_{i}{:}=x_i+\epsilon e_i t \ \ \ \ i=1,\ldots ,n. \end{aligned}$$

(3.13)

then clearly $W_i$ are wave-monogenic polynomials of degree 1. We will follow the construction of the Fueter polynomials as in [11].

For $\alpha$ as above, let $W^{\alpha }{:}=W_{1}^{k_1}W_{2}^{k_2}\dots W_{n}^{k_n}=W_{j_1}W_{j_2}\dots W_{j_k}$ where the sequence of indices $j_1,j_2,\dots ,j_k$ has the first $k_1$ indices equal 1, the next $k_2$ indices equal 2 and so on. Next we define

$$\begin{aligned} Q_{\alpha }(t,x){:}=\frac{1}{ k!}\sum _{\sigma \in Perm(S_{k})}\sigma (W^{\alpha }) {:}= \frac{1}{ k!}\sum _{\sigma \in Perm(S_{k})} W_{j_{\sigma (1)}}W_{j_{\sigma (2)}}\dots W_{j_{\sigma (k)}},\nonumber \\ \end{aligned}$$

(3.14)

where $Perm(S_m)$ denote the permutation group of m elements. This is a symmetrization process.

We see from [11] Theorem 6.2 (who proof applies to this case with minors changes) that $Q_{\alpha } \in W_k({\mathbb {R}}\times {\mathbb {R}}^n)$. Moreover it is clear that $Q(0,x)=p_{\alpha }(x)$ therefore $Q_{\alpha }=P_{\alpha }$, and it follows that any $R \in W_k({\mathbb {R}}\times {\mathbb {R}}^n)$ can be written as

$$\begin{aligned} R(t,x)=\sum _{|\alpha |=k}Q_{\alpha }(t,x)a_{\alpha } \end{aligned}$$

(3.15)

where $a_{\alpha } \in {\mathcal {C}}\,\ell (1,n)$.

4 Energy Estimates for the Dirac-Wave Operator

In this section we will see energy estimates for the Dirac-wave operator. Here n is arbitrary again. Note that we only use that u(x, t) is $C^1$ in this section. In the next result we show conservation of the energy. The proof use the Clifford algebra structure in a subtle way.

Theorem 4.1

Suppose that the initial data $\phi$ is a function of compact support and let u(x, t) be the solution of (3.1). Then $\forall t\ge 0$

$$\begin{aligned} E(t)= \int _{{\mathbb {R}}^n} \sum _{I} u^2_{I}(t,y)\, dV(y) \equiv ||\phi ||^2_{L^2( {\mathbb {R}}^n)}. \end{aligned}$$

(4.1)

Proof

We decompose u as in (2.4). Then we have that

$$\begin{aligned} ||u||^2= \sum _{I} u^2_{I}(t,y)= (u^1(t,y) \overline{u^1(t,y)})_0+(u^2(t,y) \overline{u^2(t,y)})_0. \end{aligned}$$

(4.2)

here we recall that $a_0$ is the real part of a. Therefore thanks to (2.21) and the fact that for any function f, $\overline{{\mathbb {D}}f}=- {\overline{f}}{\mathbb {D}}$ where ${\mathbb {D}}$ acts on the right in the right side of the equation.

$$\begin{aligned} \partial _t ||u||^2&=(\partial _t u^1(t,y) \overline{u^1(t,y)}+u^1(t,y) \partial _t \overline{u^1(t,y)})_0\nonumber \\&\quad +(\partial _t u^2(t,y) \overline{u^2(t,y)}+u^2(t,y) \partial _t \overline{u^2(t,y)})_0 \nonumber \\&=[ (-{\mathbb {D}}u^2(t,y) ) \overline{u^1(t,y)}+u^1(t,y) (\,\overline{u^2(t,y)}{\mathbb {D}}\,)]_0 \nonumber \\&\quad + [({\mathbb {D}}u^1(t,y)) \overline{u^2(t,y)}-u^2(t,y) (\,\overline{u^1(t,y)}{\mathbb {D}}\,) ]_0 \nonumber \\&=[({\mathbb {D}}u^1(t,y)) \overline{u^2(t,y)} +u^1(t,y) (\,\overline{u^2(t,y)}{\mathbb {D}}\,)]_0 \nonumber \\&\quad -[ ({\mathbb {D}}u^2(t,y) ) \overline{u^1(t,y)}+u^2(t,y) (\,\overline{u^1(t,y)}{\mathbb {D}}\,)]_0. \end{aligned}$$

(4.3)

We will use the following version of the diverge Theorem

$$\begin{aligned} \int _{\Omega }[ {\mathbb {D}}f\,g+f\,(g {\mathbb {D}}) ]\, dV=\int _{\partial \Omega }f\eta g \, dS. \end{aligned}$$

(4.4)

Here $\eta$ is the unit outward normal. In fact, we will only need the real component of the equation. Then we have thanks to (4.3) and (4.4) that

$$\begin{aligned} E'(t)&=\int _{{\mathbb {R}}^n}[ ({\mathbb {D}}u^1(t,y)) \overline{u^2(t,y)} +u^1(t,y) (\,\overline{u^2(t,y)}{\mathbb {D}}\,)]_0 dV(y) \nonumber \\&\quad -\int _{{\mathbb {R}}^n} [({\mathbb {D}}u^2(t,y) ) \overline{u^1(t,y)}+u^2(t,y) (\,\overline{u^1(t,y)}{\mathbb {D}}\,)]_0 dV(y) \nonumber \\&= \lim _{R\rightarrow \infty } \int _{\partial B(0,R)} [u^1(t,\omega )\eta (\omega ) \overline{u^2(t,\omega )}]_0 -[u^2(t,\omega )\eta (\omega )\overline{u^1(t,\omega )}]_0 \, dS(\omega )=0. \end{aligned}$$

(4.5)

Here we had use that u is a function of compact support. $\square$

Corollary 4.2

(Uniqueness for the Dirac-wave equation) Let $\Omega$ be a bounded open set with smooth boundary in ${{\mathbb {R}}}^n$ and let $T> 0$. Then there exist at most one function $u\in {{C}}^1((0, T) \times \Omega ,{\mathcal {C}}\,\ell (1,n) \, )$ solving

$$\begin{aligned} \begin{array}{l} {\mathbb {D}}_w u(t,x)=0 \ \ \ \ in \ \ (0, T) \times \Omega , \\ u(0,x)=\phi (x) \ \ \ \ x \in \Omega , \\ u(t,y)=g(t,y) \ \ \ \ in \ \ (0, T) \times \partial \Omega \end{array} \end{aligned}$$

(4.6)

for some function g.

Proof

if u and ${\hat{u}}$ are solutions of (4.6) let $w=u-{\hat{u}}$ then we proceed to compute the energy of w as above. We have no boundary terms. Therefore $E(t)\equiv 0$ and $w\equiv 0$. $\square$

Corollary 4.3

(Finite propagation speed) Suppose u(t, x) is a solution of (3.1). Fix $x_0 \in {\mathbb {R}}^n$ and $t_0> 0$. If $\phi (x)\equiv 0$ on $B(x_0,t_0)$ then $u(t,x)\equiv 0$ within the cone $C=\{\, (t,x)\, |\, 0 \le t \le t_0, \, \Vert x-x_0\Vert \le t_0 -t \, \}$.

Proof

If

$$\begin{aligned} E(t)= \int _{B(x_0,t_0-t)} \sum _{I} u^2_{I}(t,y)\, dV(y) \end{aligned}$$

(4.7)

for $0 \le t \le t_0$ then using the notation and the proof in Theorem (4.1) we had that

$$\begin{aligned} E'(t)&=\int _{\partial B(x_0,t_0-t)}[u^1(t,\omega )\eta (\omega ) \overline{u^2(t,\omega )}-u^2(t,\omega )\eta (\omega )\overline{u^1(t,\omega )}]_0 \, dS(\omega )\nonumber \\&\quad -\int _{\partial B(x_0,t_0-t)} \sum _{I} u^2_{I}(t,\omega )\,dS(\omega ) \end{aligned}$$

(4.8)

Here $\eta$ is the unit outward normal. Then on the sphere $\partial B(x_0,t_0-t)$

$$\begin{aligned} ||u^1\eta \overline{u^2}-u^2\eta \overline{u^1}||\le 2 ||u^1||\,||u^2|| \le ||u^1||^2+||u^2||^2=||u||^2. \end{aligned}$$

(4.9)

Hence $E'(t)\le 0$ for $0 \le t \le t_0$ and $E(0)=0$ implies that $E(t)\equiv 0$. Therefore $u(t,x)\equiv 0$ within C. $\square$

5 Dirac Wave Conjugates Functions

In this section we define the Dirac-wave conjugate to a wave function when $n=3$. Again the other cases are analogous.

For starter given a scalar wave function $u_{0}$ we would like to find a Clifford-valued function u with real part $u_{0}$ which satisfaces the Dirac-wave operator. This is a wave-monogenic extension of $u_{0}$. The Dirac-wave conjugate function will be then $u-u_{0}$. Also from this, as it is done in the complex analysis case, we would like to define a Hilbert-wave transform as the limit as t go to zero of the Dirac-wave conjugate function. Actually more is true. We can give four wave functions $u_i(t,x)$ each one with initial data $\phi _i (x)$ and with initial velocity $\psi _i(x)$, $i=0,1,2,3$. It will be possible to extend them to a wave-monogenic function if we take

$$\begin{aligned} u^1(t,x)=u_0(t,x)+e_1e_2u_1(t,x)+e_1e_3u_2(t,x)+e_2e_3 u_3(t,x). \end{aligned}$$

(5.1)

Note that we need to take functions which are smooth and of compact support in order to have a classical solution to the wave operator.

Lemma 5.1

Let $\phi _i(x),\psi _i(x) \in C^{\infty } ({\mathbb {R}}^3)\$ with compact support for $i=0,1,2,3$. Suppose $u_i$ is the (scalar) wave function with initial data $\phi _i$ and with initial velocity $\psi _i$ and let $u^1(t,x)$ be as in (5.1) then there exist a wave-monogenic function u with values in ${\mathcal {C}}\,\ell ^{+}(1,3)$ which extend $u^1(t,x)$. In particular, any scalar wave function can extended to a wave-monogenic function.

Proof

If we take $\Phi (x)=u^1(0,x)$ as initial value in (3.1), its solution

$$\begin{aligned} v_1(t,x)=\partial _t (\, ({\mathcal {R}} \Phi )(t,x) \, )+{\mathcal {R}}\big ( \epsilon {\mathbb {D}} \Phi \big ) (t,x), \end{aligned}$$

(5.2)

will be wave-monogenic with initial value $\Phi (x)$. Note that $\epsilon {\mathbb {D}} \Phi (x)$ takes values in the subspace generated by $\{ \epsilon e_1,\epsilon e_2,\epsilon e_3, \epsilon e_1e_2e_3 \}$. We call this subspace M.

Now let $\Psi (x)=\partial _t u^1(0,x)$ and consider the Teodorescu transform of $\Psi$, $T(\Psi )$ which is a function of compact support. Note that $T(\Psi (x))\epsilon$ takes values also in the subspace M. Therefore the solution to (3.1) with initial data $T(\Psi )\epsilon$ is given by

$$\begin{aligned} v_2(t,x)= & {} \partial _t ( {\mathcal {R}}(T \Psi \epsilon ) (t,x) )+{\mathcal {R}}\big ( \epsilon {\mathbb {D}} (T\Psi \epsilon ) \big ) (t,x)\nonumber \\= & {} \partial _t ( {\mathcal {R}}(T \Psi \epsilon ) (t,x) )+{\mathcal {R}}( \Psi ) (t,x), \end{aligned}$$

(5.3)

we had use (2.15).

If we take $u= v_1+v_2$ then u is wave-monogenic and extend $u^1$. $\square$

From the construction above we see that we can define the wave-conjugate of $u^1$ as

$$\begin{aligned} \widetilde{u^1(t,x)}{:}= {\mathcal {R}}\big ( \epsilon {\mathbb {D}} \Phi \big ) (t,x)+\partial _t ( {\mathcal {R}}(T \Psi \epsilon ) (t,x) ) \end{aligned}$$

(5.4)

Note that the corresponding wave Hilbert transform will be the Teodorescu transform $T \Psi \epsilon$.

The next corollary will give a decomposition of a wave function with values in the Clifford algebra as the sum of a wave-monogenic function and an anti wave-monogenic function.

Corollary 5.2

Suppose u is a solution with compact support of the BVP for the wave function with values in ${\mathcal {C}}\,\ell ^{+}(1,3)$ then there exist Clifford values functions $v_1$ and $v_2$ such that ${\mathbb {D}}_w v_1=0$, $\widetilde{{\mathbb {D}}_w}(v_2)=0$ and $u=v_1 +v_2$. The decomposition is unique module monogenic solutions.

Proof

Decompose $u=u^1+\epsilon u^2$ as in (2.4). Then using lemma (5.1) with $u^1$ and $u^2$ and (2.1) (ii), there exist wave monogenic functions $v_1$ and $w_1$ which extend $u^1$ and $u^2$ respectively. Let $v_2=\epsilon w_1$ then thanks to (2.19) $v_2$ is an anti wave-monogenic function. Then $u=v_1 +v_2$. Next if ${\mathbb {D}}_w u=\widetilde{{\mathbb {D}}_w}(u)$ it easily follows that $\partial _t u=0$. $\square$