A Cauchy-Kowalevski theorem for inframonogenic functions

TL;DR

This paper establishes a Cauchy-Kowalevski theorem specifically for inframonogenic functions, which satisfy a particular differential system, expanding the theoretical understanding of these functions.

Contribution

The paper introduces a Cauchy-Kowalevski theorem tailored for inframonogenic functions, providing new theoretical insights into their properties.

Findings

Proves a Cauchy-Kowalevski theorem for inframonogenic functions

Extends classical PDE theory to a new class of functions

Provides foundational results for further research in this area

Abstract

In this paper we prove a Cauchy-Kowalevski theorem for the functions satisfying the system DfD=0 (called inframonogenic functions).

Equations

$$a=\sum _A{a}_A{e}_A,a_A\in \mathbb{R},$$

$$\mathbb{R}_{0,m}^{(k)}=\biggl{\{}a\in\mathbb{R}_{0,m}:\;a=\sum_{|A|=k}a_{A}e_{A},\quad a_{A}\in\mathbb{R}\biggr{\}}.$$

$$\overline{a}=\sum _A{a}_A{\overline{e}}_A,$$

$$|a{|}^2=[a\overline{a}{]}_0=\sum _A{a}_A^2.$$

$${\Delta }_x=\sum _{j=0}^m{\partial }_{x_j}^2={\partial }_x{\overline{\partial }}_x={\overline{\partial }}_x{\partial }_x,$$

$$F(x){|}_{x_0=0}$$

$${\partial }_{x_0}F(x){|}_{x_0=0}$$

$$F(x)=\sum _{n=0}^{\infty }{x}_0^n{A}_n(\underline{x}).$$

$$\partial_{x}\left(x_{0}^{n}A_{n}\right)\partial_{x}=n(n-1)x_{0}^{n-2}A_{n}+nx_{0}^{n-1}\big{(}\partial_{\underline{x}}A_{n}+A_{n}\partial_{\underline{x}}\big{)}+x_{0}^{n}\partial_{\underline{x}}A_{n}\partial_{\underline{x}}.$$

$$\partial_{x}F\partial_{x}=\sum_{n=0}^{\infty}x_{0}^{n}\Big{(}(n+2)(n+1)A_{n+2}+(n+1)\big{(}\partial_{\underline{x}}A_{n+1}+A_{n+1}\partial_{\underline{x}}\big{)}+\partial_{\underline{x}}A_{n}\partial_{\underline{x}}\Big{)}.$$

$$A_{n+2}=-\frac{1}{(n+2)(n+1)}\Big{(}(n+1)\big{(}\partial_{\underline{x}}A_{n+1}+A_{n+1}\partial_{\underline{x}}\big{)}+\partial_{\underline{x}}A_{n}\partial_{\underline{x}}\Big{)},\quad n\geq 0.$$

$$A_n=\frac{(-1{)}^{n+1}}{n!}\left(\sum _{j=0}^{n-2}{\partial }_{\underline{x}}^{n-j-1}{A}_0{\partial }_{\underline{x}}^{j+1}+\sum _{j=0}^{n-1}{\partial }_{\underline{x}}^{n-j-1}{A}_1{\partial }_{\underline{x}}^j\right),n\ge 2.$$

$$\left|\partial_{\underline{x}}^{n-j}A_{s}(\underline{x})\partial_{\underline{x}}^{j}\right|\leq M(\underline{y})\frac{n!}{R^{n}(\underline{y})},\;\;\underline{x}\in B\big{(}\underline{y},R(\underline{y})\big{)},\;\;j=0,\dots,n,\;\;s=0,1.$$

$$\left|A_{n}(\underline{x})\right|\leq M(\underline{y})\frac{n+R(\underline{y})-1}{R^{n}(\underline{y})},\quad\underline{x}\in B\big{(}\underline{y},R(\underline{y})\big{)},$$

$$\widetilde{\Omega}=\bigcup_{\underline{y}\in\underline{\Omega}}\left(-R(\underline{y}),R(\underline{y})\right)\times B\big{(}\underline{y},R(\underline{y})\big{)}.$$

$$\begin{gathered} \mathsf{CK}[A_0,A_1](x)=A_0(\underline{x})+x_0{A}_1(\underline{x}) \\ -\sum _{n=2}^{\infty }\frac{(-x_0{)}^n}{n!}\left(\sum _{j=0}^{n-2}{\partial }_{\underline{x}}^{n-j-1}{A}_0(\underline{x}){\partial }_{\underline{x}}^{j+1}+\sum _{j=0}^{n-1}{\partial }_{\underline{x}}^{n-j-1}{A}_1(\underline{x}){\partial }_{\underline{x}}^j\right) \end{gathered}$$

$$\mathsf{CK}[A_0,-{\partial }_{\underline{x}}{A}_0](x)=\sum _{n=0}^{\infty }\frac{(-x_0{)}^n}{n!}{\partial }_{\underline{x}}^n{A}_0(\underline{x}),$$

$$\begin{gathered} \mathsf{CK}[P_k,P_{k-1}](x)=P_k(\underline{x})+x_0{P}_{k-1}(\underline{x}) \\ -\sum _{n=2}^k\frac{(-x_0{)}^n}{n!}\left(\sum _{j=0}^{n-2}{\partial }_{\underline{x}}^{n-j-1}{P}_k(\underline{x}){\partial }_{\underline{x}}^{j+1}+\sum _{j=0}^{n-1}{\partial }_{\underline{x}}^{n-j-1}{P}_{k-1}(\underline{x}){\partial }_{\underline{x}}^j\right), \end{gathered}$$

$$P_k(\underline{x})=P_k({\partial }_{\underline{u}})\frac{{⟨\underline{x},\underline{u}⟩}^k}{k!},P_k(\underline{x})\in \mathsf{P}(k),$$

$$\mathsf{CK}\big{[}\left\langle\underline{x},\underline{u}\right\rangle^{k}e_{A},0\big{]}(x)=\left\langle\underline{x},\underline{u}\right\rangle^{k}e_{A}\\ -\sum_{n=2}^{k}\binom{k}{n}(-x_{0})^{n}\left\langle\underline{x},\underline{u}\right\rangle^{k-n}\left(\sum_{j=0}^{n-2}\underline{u}^{n-j-1}e_{A}\underline{u}^{j+1}\right),$$

$$\mathsf{CK}\big{[}0,\left\langle\underline{x},\underline{u}\right\rangle^{k-1}e_{A}\big{]}(x)=x_{0}\left\langle\underline{x},\underline{u}\right\rangle^{k-1}e_{A}\\ -\frac{1}{k}\sum_{n=2}^{k}\binom{k}{n}(-x_{0})^{n}\left\langle\underline{x},\underline{u}\right\rangle^{k-n}\left(\sum_{j=0}^{n-1}\underline{u}^{n-j-1}e_{A}\underline{u}^{j}\right).$$

Full text

Keywords: Inframonogenic functions; Cauchy-Kowalevski theorem.

Mathematics Subject Classification: 30G35.

1 Introduction

Let $\mathbb{R}_{0,m}$ be the $2^{m}$-dimensional real Clifford algebra constructed over the orthonormal basis $(e_{1},\ldots,e_{m})$ of the Euclidean space $\mathbb{R}^{m}$ (see [3]). The multiplication in $\mathbb{R}_{0,m}$ is determined by the relations $e_{j}e_{k}+e_{k}e_{j}=-2\delta_{jk}$, $j,k=1,\dots,m$, where $\delta_{jk}$ is the Kronecker delta. A general element of $\mathbb{R}_{0,m}$ is of the form

[TABLE]

where for $A=\{j_{1},\dots,j_{k}\}\subset\{1,\dots,m\}$, $j_{1}<\cdots<j_{k}$, $e_{A}=e_{j_{1}}\dots e_{j_{k}}$. For the empty set $\emptyset$, we put $e_{\emptyset}=1$, the latter being the identity element.

Notice that any $a\in\mathbb{R}_{0,m}$ may also be written as $a=\sum_{k=0}^{m}[a]_{k}$ where $[a]_{k}$ is the projection of $a$ on $\mathbb{R}_{0,m}^{(k)}$. Here $\mathbb{R}_{0,m}^{(k)}$ denotes the subspace of $k$-vectors defined by

[TABLE]

Observe that $\mathbb{R}^{m+1}$ may be naturally identified with $\mathbb{R}_{0,m}^{(0)}\oplus\mathbb{R}_{0,m}^{(1)}$ by associating to any element $(x_{0},x_{1},\ldots,x_{m})\in\mathbb{R}^{m+1}$ the “paravector” $x=x_{0}+\underline{x}=x_{0}+\sum_{j=1}^{m}x_{j}e_{j}$.

Conjugation in $\mathbb{R}_{0,m}$ is given by

[TABLE]

where $\overline{e}_{A}=\overline{e}_{j_{k}}\dots\overline{e}_{j_{1}}$, $\overline{e}_{j}=-e_{j}$, $j=1,\dots,m$. One easily checks that $\overline{ab}=\overline{b}\overline{a}$ for any $a,b\in\mathbb{R}_{0,m}$. Moreover, by means of the conjugation a norm $|a|$ may be defined for each $a\in\mathbb{R}_{0,m}$ by putting

[TABLE]

Let us denote by $\partial_{x}=\partial_{x_{0}}+\partial_{\underline{x}}=\partial_{x_{0}}+\sum_{j=1}^{m}e_{j}\partial_{x_{j}}$ the generalized Cauchy-Riemann operator and let $\Omega$ be an open set of $\mathbb{R}^{m+1}$. According to [11], an $\mathbb{R}_{0,m}$-valued function $f\in C^{2}(\Omega)$ is called an inframonogenic function in $\Omega$ if and only if it fulfills in $\Omega$ the “sandwich” equation $\partial_{x}f\partial_{x}=0$.

It is obvious that monogenic functions (i.e. null-solutions of $\partial_{x}$) are inframonogenic. At this point it is worth remarking that the monogenic functions are the central object of study in Clifford analysis (see [2, 4, 5, 7, 8, 9, 10, 14]). Furthermore, the concept of monogenicity of a function may be seen as the higher dimensional counterpart of holomorphy in the complex plane.

Moreover, as

[TABLE]

every inframonogenic function $f\in C^{4}(\Omega)$ satisfies in $\Omega$ the biharmonic equation $\Delta_{x}^{2}f=0$ (see e.g. [1, 6, 12, 15]).

This paper is intended to study the following Cauchy-type problem for the inframonogenic functions. Given the functions $A_{0}(\underline{x})$ and $A_{1}(\underline{x})$ analytic in an open and connected set $\underline{\Omega}\subset\mathbb{R}^{m}$, find a function $F(x)$ inframonogenic in some open neighbourhood $\widetilde{\Omega}$ of $\underline{\Omega}$ in $\mathbb{R}^{m+1}$ which satisfies

[TABLE]

2 Cauchy-type problem for inframonogenic functions

Consider the formal series

[TABLE]

It is clear that $F$ satisfies conditions (1) and (2). We also see at once that

[TABLE]

We thus get

[TABLE]

From the above it follows that $F$ is inframonogenic if and only if the functions $A_{n}$ satisfy the recurrence relation

[TABLE]

It may be easily proved by induction that

[TABLE]

We now proceed to examine the convergence of the series (3) with the functions $A_{n}$ ($n\geq 2$) given by (4). Let $\underline{y}$ be an arbitrary point in $\underline{\Omega}$. Then there exist a ball B\big{(}\underline{y},R(\underline{y})\big{)} of radius $R(\underline{y})$ centered at $\underline{y}$ and a positive constant $M(\underline{y})$, such that

[TABLE]

It follows that

[TABLE]

and therefore the series (3) converges normally in

[TABLE]

Note that $\widetilde{\Omega}$ is a $x_{0}$-normal open neighbourhood of $\underline{\Omega}$ in $\mathbb{R}^{m+1}$, i.e. for each $x\in\widetilde{\Omega}$ the line segment $\{x+t:\,t\in\mathbb{R}\}\cap\widetilde{\Omega}$ is connected and contains one point in $\underline{\Omega}$.

We thus have proved the following.

Theorem 1

The function $\mathsf{CK}[A_{0},A_{1}]$ given by

[TABLE]

is inframonogenic in a $x_{0}$-normal open neighbourhood of $\,\underline{\Omega}$ in $\mathbb{R}^{m+1}$ and satisfies conditions (1)-(2).

It is worth noting that if in particular $A_{1}(\underline{x})=-\partial_{\underline{x}}A_{0}(\underline{x})$, then

[TABLE]

which is nothing else but the left monogenic extension (or CK-extension) of $A_{0}(\underline{x})$. Similarly, it is easy to see that $\mathsf{CK}[A_{0},-A_{0}\partial_{\underline{x}}](x)$ yields the right monogenic extension of $A_{0}(\underline{x})$ (see [2, 5, 13, 16, 17]).

Let $\mathsf{P}(k)$ ($k\in\mathbb{N}_{0}$ fixed) denote the set of all $\mathbb{R}_{0,m}$-valued homogeneous polynomials of degree $k$ in $\mathbb{R}^{m}$. Let us now take $A_{0}(\underline{x})=P_{k}(\underline{x})\in\mathsf{P}(k)$ and $A_{1}(\underline{x})=P_{k-1}(\underline{x})\in\mathsf{P}(k-1)$. Clearly,

[TABLE]

since the other terms in the series (5) vanish. Moreover, we can also claim that $\mathsf{CK}[P_{k},P_{k-1}](x)$ is a homogeneous inframonogenic polynomial of degree $k$ in $\mathbb{R}^{m+1}$.

Conversely, if $P_{k}(x)$ is a homogeneous inframonogenic polynomial of degree $k$ in $\mathbb{R}^{m+1}$, then $P_{k}(x)|_{x_{0}=0}\in\mathsf{P}(k)$, $\partial_{x_{0}}P_{k}(x)|_{x_{0}=0}\in\mathsf{P}(k-1)$ and obviously $\mathsf{CK}[P_{k}|_{x_{0}=0},\partial_{x_{0}}P_{k}|_{x_{0}=0}](x)=P_{k}(x)$.

Call $\mathsf{I}(k)$ the set of all homogeneous inframonogenic polynomials of degree $k$ in $\mathbb{R}^{m+1}$. Then $\mathsf{CK}[.,.]$ establishes a bijection between $\mathsf{P}(k)\times\mathsf{P}(k-1)$ and $\mathsf{I}(k)$.

It is easy to check that

[TABLE]

where $P_{k}(\partial_{\underline{u}})$ is the differential operator obtained by replacing in $P_{k}(\underline{u})$ each variable $u_{j}$ by $\partial_{u_{j}}$. Therefore, in order to characterize $\mathsf{I}(k)$, it suffices to calculate \mathsf{CK}\big{[}\left\langle\underline{x},\underline{u}\right\rangle^{k}e_{A},0\big{]} and \mathsf{CK}\big{[}0,\left\langle\underline{x},\underline{u}\right\rangle^{k-1}e_{A}\big{]} with $\underline{u}\in\mathbb{R}^{m}$.

A simple computation shows that

[TABLE]

[TABLE]

Acknowledgments

The second author was supported by a Post-Doctoral Grant of Fundação para a Ciência e a Tecnologia, Portugal (grant number: SFRH/BPD/45260/2008).