[Paper Review] A Cauchy-Kowalevski theorem for inframonogenic functions

Helmuth R. Malonek, Dixan Peña Peña+1 more|ArXiv.org|Nov 4, 2009

Algebraic and Geometric Analysis16 references21 citations

TL;DR

This paper establishes a Cauchy-Kowalevski-type theorem for inframonogenic functions in Clifford analysis, proving the existence of a unique solution to a Cauchy-type initial value problem in a $x_0$-normal neighborhood. The solution is constructed via a convergent power series in $x_0$ with coefficients recursively determined by differential operators acting on analytic initial data $A_0$ and $A_1$.

ABSTRACT

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

Motivation & Objective

• To establish a Cauchy-Kowalevski-type existence and uniqueness result for inframonogenic functions in higher-dimensional Clifford analysis.
• To solve the initial value problem where an inframonogenic function and its $x_0$-derivative are prescribed on the hyperplane $x_0 = 0$.
• To characterize the solution as a convergent power series in $x_0$ with coefficients derived from differential operators acting on analytic initial data.
• To show that the solution operator, denoted $\mathsf{CK}[A_0, A_1]$, provides a bijection between initial data in $\mathsf{P}(k) \times \mathsf{P}(k-1)$ and homogeneous inframonogenic polynomials of degree $k$ in $\mathbb{R}^{m+1}$.

Proposed method

• Construct a formal power series solution $F(x) = \sum_{n=0}^\infty x_0^n A_n(\underline{x})$ in $x_0$ with coefficients $A_n$ to satisfy the initial conditions $F|_{x_0=0} = A_0(\underline{x})$ and $\partial_{x_0}F|_{x_0=0} = A_1(\underline{x})$.
• Derive a recurrence relation for $A_n$ by requiring $\partial_x F \partial_x = 0$, leading to $A_{n+2} = -\frac{1}{(n+2)(n+1)}\left((n+1)(\partial_{\underline{x}}A_{n+1} + A_{n+1}\partial_{\underline{x}}) + \partial_{\underline{x}}A_n\partial_{\underline{x}}\right)$.
• Prove convergence of the series in a $x_0$-normal open neighborhood $\widetilde{\Omega}$ of $\underline{\Omega} \subset \mathbb{R}^m$ using majorant estimates based on analyticity of $A_0$ and $A_1$.
• Define the solution operator $\mathsf{CK}[A_0, A_1]$ explicitly as a series involving iterated applications of $\partial_{\underline{x}}$ and $\partial_{\underline{x}}$ on $A_0$ and $A_1$.
• Establish a bijection between homogeneous inframonogenic polynomials of degree $k$ in $\mathbb{R}^{m+1}$ and pairs of homogeneous polynomials of degree $k$ and $k-1$ in $\mathbb{R}^m$ via the $\mathsf{CK}$ operator.
• Explicitly compute $\mathsf{CK}$ for monomials $\langle\underline{x},\underline{u}\rangle^k e_A$ and $\langle\underline{x},\underline{u}\rangle^{k-1} e_A$ to characterize the full space $\mathsf{I}(k)$.

Experimental results

Research questions

• RQ1 Does a solution exist to the Cauchy problem for inframonogenic functions with analytic initial data on $x_0 = 0$?
• RQ2 Can the solution be expressed as a convergent power series in $x_0$ with coefficients determined by differential operators on the initial data?
• RQ3 Is the solution operator $\mathsf{CK}[A_0, A_1]$ a bijection between initial data and homogeneous inframonogenic polynomials of degree $k$?
• RQ4 How does the $\mathsf{CK}$ operator relate to the classical monogenic extension when $A_1 = -\partial_{\underline{x}}A_0$?
• RQ5 What is the explicit form of the $\mathsf{CK}$ extension for homogeneous polynomial initial data?

Key findings

• The solution $\mathsf{CK}[A_0, A_1]$ is an inframonogenic function in a $x_0$-normal open neighborhood $\widetilde{\Omega}$ of $\underline{\Omega}$ in $\mathbb{R}^{m+1}$, satisfying the initial conditions $F|_{x_0=0} = A_0(\underline{x})$ and $\partial_{x_0}F|_{x_0=0} = A_1(\underline{x})$.
• The series solution converges normally in $\widetilde{\Omega} = \bigcup_{\underline{y} \in \underline{\Omega}} (-R(\underline{y}), R(\underline{y})) \times B(\underline{y}, R(\underline{y}))$, where $R(\underline{y})$ is the radius of convergence of the initial data.
• When $A_1 = -\partial_{\underline{x}}A_0$, the $\mathsf{CK}[A_0, -\partial_{\underline{x}}A_0]$ reduces to the left monogenic extension $\sum_{n=0}^\infty \frac{(-x_0)^n}{n!} \partial_{\underline{x}}^n A_0(\underline{x})$.
• The $\mathsf{CK}$ operator establishes a bijection between $\mathsf{P}(k) \times \mathsf{P}(k-1)$ and the space $\mathsf{I}(k)$ of homogeneous inframonogenic polynomials of degree $k$ in $\mathbb{R}^{m+1}$.
• For initial data $P_k(\underline{x}) \in \mathsf{P}(k)$ and $P_{k-1}(\underline{x}) \in \mathsf{P}(k-1)$, the solution $\mathsf{CK}[P_k, P_{k-1}]$ is a homogeneous inframonogenic polynomial of degree $k$ in $\mathbb{R}^{m+1}$.
• Explicit formulas are derived for $\mathsf{CK}[\langle\underline{x},\underline{u}\rangle^k e_A, 0]$ and $\mathsf{CK}[0, \langle\underline{x},\underline{u}\rangle^{k-1} e_A]$, involving binomial coefficients and iterated Clifford products of $\underline{u}$.