\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{amssymb} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2010(2010), No. 76, pp. 1--9.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2010 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2010/76\hfil Solution of a PDE] {Solutions of a partial differential equation related to the oplus operator} \author[W. Satsanit\hfil EJDE-2010/76\hfilneg] {Wanchak Satsanit} \address{Wanchak Satsanit \newline Department of Mathematics\\ Faculty of Science, Maejo University\\ Chiang Mai, 50290 Thailand} \email{aunphue@live.com} \thanks{Submitted April 8, 2010. Published June 8, 2010.} \subjclass[2000]{46F10, 46F12} \keywords{Ultra-hyperbolic kernel; diamond operator; tempered distribution} \begin{abstract} In this article, we consider the equation $$\oplus^ku(x)=\sum^{m}_{r=0}c_{r}\oplus^{r}\delta$$ where $\oplus^k$ is the operator iterated $k$ times and defined by $$\oplus^k=\Big(\Big(\sum^p_{i=1}\frac{\partial^2}{\partial x^2_i}\Big)^{4}-\Big(\sum^{p+q}_{j=p+1}\frac{\partial^2}{\partial x^2_j}\Big)^{4}\Big)^k,$$ where $p+q=n$, $x=(x_1,x_2,\dots,x_n)$ is in the $n$-dimensional Euclidian space $\mathbb{R}^n$, $c_{r}$ is a constant, $\delta$ is the Dirac-delta distribution, $\oplus^{0}\delta=\delta$, and $k=0,1,2,3,\dots$. It is shown that, depending on the relationship between $k$ and $m$, the solution to this equation can be ordinary functions, tempered distributions, or singular distributions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \section{Introduction} The diamond operator, iterated $k$ times, was studied by Kananthai \cite{k1}, and is defined by $$\label{1.1} \diamondsuit^k=\Big(\Big(\sum^p_{i=1}\frac{\partial^2}{\partial x^2_i}\Big)^2 -\Big(\sum^{p+q}_{j=p+1}\frac{\partial^2}{\partial x^2_j}\Big)^2\Big)^k,\quad p+q=n,$$ where $n$ is the dimension of the space $\mathbb{R}^n$, $x=(x_1,x_2,\dots,x_n)\in \mathbb{R}^n$, and $k$ is a nonnegative integer. This operator can be expressed as $$\label{1.2} \diamondsuit^k=\Delta^k\square^k=\square^k\Delta^k$$ where $\Delta^k$ is the Laplacian operator iterated $k$ times, defined by $$\label{1.3} \Delta^k = \Big(\frac{\partial^2}{\partial x^2_1}+\frac{\partial^2}{\partial x^2_2}+\dots+\frac{\partial^2}{\partial x^2_n}\Big)^k,$$ and $\square^k$ is the Ultra-hyperbolic operator iterated $k$ times, defined by $$\label{1.4} \square^k = \Big(\frac{\partial^2}{\partial x_1^2}+ \frac{\partial^2}{\partial x_2^2}+\dots+\frac{\partial^2}{\partial x_p^2}-\frac{\partial^2}{\partial x_{p+1}^2}-\frac{\partial^2}{\partial x_{p+2}^2}-\dots-\frac{\partial^2}{\partial x_{p+q}^2}\Big)^k.$$ Kananthai \cite{k1} showed that the convolution $$u(x)=(-1)^kR^{e}_{2k}(x)\ast R^{H}_{2k}(x)$$ is a unique elementary solution of the operator $\diamondsuit^k$, where $R^{e}_{2k}(x)$ and $R^{H}_{2k}(x)$ are defined by \eqref{2.5} and \eqref{2.2} with $\alpha=2k$ respectively; that is, $$\label{1.5} \diamondsuit^k\left((-1)^kR^{e}_{2k}(x)\ast R^{H}_{2k}(x)\right)=\delta\,.$$ Satsanit \cite{s2} introduced the $\circledcirc^k$ operator, defined by $\circledcirc^k=\Big(\big(\sum^p_{i=1}\frac{\partial^2}{\partial x^2_i}\Big)^2 +\Big(\sum^{p+q}_{j=p+1}\frac{\partial^2}{\partial x^2_j}\Big)^2\Big)^k.$ From \eqref{1.3} and \eqref{1.4}, we obtain \label{1.6} \begin{aligned} \circledcirc^k &= \Big(\Big(\sum^p_{i=1}\frac{\partial^2}{\partial x^2_i}\Big)^2 +\Big(\sum^{p+q}_{j=p+1}\frac{\partial^2}{\partial x^2_j}\Big)^2\Big)^k \\ &= \Big(\Big(\frac{\Delta+\square}{2}\Big)^{2} +\Big(\frac{\Delta-\square}{2}\Big)^{2}\Big)^k \\ &= \Big(\frac{\Delta^2+\square^2}{2}\Big)^k . \end{aligned} The $\oplus^k$ operator has been studied by Kananthai, Suantai and Longani \cite{k3}, and can be expressed in the form $$\label{1.7} \oplus^k=\Big[\Big(\sum^p_{i=1}\frac{\partial^2}{\partial x_i^2}\Big)^2 -\Big(\sum^{p+q}_{j=p+1}\frac{\partial^2}{\partial x_j^2}\Big)^2\Big]^k\cdot\Big[\Big(\sum^p_{i=1}\frac{\partial^2}{\partial x_i^2}\Big)^2 +\Big(\sum^{p+q}_{j=p+1}\frac{\partial^2}{\partial x_j^2}\Big)^2\Big]^k$$ Thus, \eqref{1.7} can be written as $$\label{1.8} \oplus^k=\diamondsuit^k\circledcirc^k,$$ where $\diamondsuit^k$ and $\circledcirc^k$ are defined by \eqref{1.1}, \eqref{1.6} respectively. The purpose of this article, is finding the solution to the equation $$\label{1.9} \oplus^ku(x)=\sum^{m}_{r=0}c_{r}\oplus^{r}\delta$$ by using convolutions of the generalized function. It is also shown that the type of solution to \eqref{1.9} depends on the relationship between $k$ and $m$, according to the following cases: \begin{itemize} \item[(1)] If $m 0$ and $u > 0 \}$ be the interior of a forward cone and let $\overline{\Gamma}_+$ denote its closure. For any complex number $\alpha$, define the function $$\label{2.2} R_\alpha^H (\upsilon) = \begin{cases} \frac{\upsilon^{(\alpha - n)/2}}{K_n (\alpha)}, &\text{for } x \in \Gamma_+, \\ 0, &\text{for }x \not\in \Gamma_+, \end{cases}$$ where $$\label{2.3} K_n (\alpha) = \frac{\pi^{\frac{n-1}{2}} \Gamma ( \frac{2 + \alpha - n}{2} ) \Gamma ( \frac{1 - \alpha}{2}) \Gamma (\alpha)}{\Gamma ( \frac{2 + \alpha - p}{2} ) \Gamma ( \frac{p - \alpha}{2})}.$$ The function $R_{\alpha}^H(\upsilon)$ was introduced by Nozaki \cite[p. 72]{n1} and is called the Ultra-hyperbolic kernel of Marcel Riesz. \end{definition} It is well known that $R_\alpha^H(\upsilon)$ is an ordinary function if $\mathop{\rm Re }(\alpha) \ge n$ and is a distribution of $\alpha$ if $\mathop{\rm Re }(\alpha) < n$. Let $\mathop{\rm supp}R_\alpha^H (\upsilon)$ denote the support of $R_\alpha^H (\upsilon)$ and suppose $\mathop{\rm supp}R_\alpha^H (\upsilon)\subset \bar{\Gamma}_+$, that is $\mathop{\rm supp}R_\alpha^H (\upsilon)$ is compact. From Trione \cite[p. 11]{t2}, $R^{H}_{2k}(\upsilon)$ is an elementary solution of the operator $\square^k$; that is, $$\label{2.4} \square^kR^{H}_{2k}(\upsilon)=\delta(x)\,.$$ \begin{definition} \label{def2.2} \rm Let $x = ( x_1, x_2, \dots, x_n )$ and $|x| = (x_1^2 + x_2^2 + \dots + x_n^2)^{1/2}$. The elliptic kernel of Marcel Riesz and is defined as $$\label{2.5} R_\alpha^e (x) =\frac{ |x|^{\alpha-n}}{W_n(\alpha)}$$ where $$\label{2.6} W_n(\alpha) =\frac{ \pi^{\frac{n}{2}}2^{\alpha}\Gamma\left(\frac{\alpha}{2}\right)} {\Gamma\left(\frac{n-\alpha}{2}\right)},$$ $\alpha$ is a complex parameter, and $n$ is the dimension of $\mathbb{R}^n$. \end{definition} It can be shown that $R^{e}_{-2k}(x)=(-1)^k\Delta^k\delta(x)$ where $\Delta^k$ is defined by \eqref{1.3}. It follows that $R^{e}_{0}(x)=\delta(x)$, \cite[p. 118]{k1}. Moreover, $(-1)^kR^{e}_{2k}(x)$ is an elementary solution of the operator $\Delta^k$ \cite[Lemma 2.4]{k1}; that is, $$\label{2.7} \Delta^k((-1)^kR^{e}_{2k}(x)=\delta(x)\,.$$ \begin{lemma} \label{lem2.1} The functions $R^{H}_{2k}(\upsilon)$ and $(-1)^kR^{e}_{2k}(x)$ are the elementary solutions of the operators $\square^k$ and $\Delta^k$, defined by \eqref{1.4} and \eqref{1.3} respectively. The function $R^{H}_{2k}(\upsilon)$ is defined by \eqref{2.2} with $\alpha=2k$, and $R^{e}_{2k}(x)$ is defined by \eqref{2.5} with $\alpha=2k$. \end{lemma} \begin{proof} We need to show that $\square^kR^{H}_{2k}(\upsilon)=\delta(x)$ which is done in \cite[Lemma 2.4]{t2}. Also we need to show that $\Delta^k((-1)^kR^{e}_{2k}(x)=\delta(x)$. which is done in \cite[p. 31]{k1}. \end{proof} \begin{lemma} \label{lem2.2} The convolution $R^{H}_{2k}(\upsilon)\ast (-1)^kR^{e}_{2k}(x)$ is an elementary solution of the operator $\diamondsuit^k$ iterated $k$ as defined by \eqref{1.1}. \end{lemma} For the proof of the above lemma see \cite[p. 33]{k1}. \begin{lemma} \label{lem2.3} The functions $R^{H}_{\alpha}(x)$ and $R^{e}_{\alpha}(x)$ defined by \eqref{2.2} and \eqref{2.5} respectively, for $Re(\alpha)$, are homogeneous distributions of order $\alpha-n$ and also a tempered distributions. \end{lemma} \begin{proof} Since $R^{H}_{\alpha}(x)$ and $R^{e}_{\alpha}(x)$ satisfy the Euler equation, \begin{gather*} (\alpha-n)R^{H}_{\alpha}(x)=\sum^{n}_{i=1}x_{i}\frac {\partial}{\partial x_{i}}R^{H}_{\alpha}(x), \\ (\alpha-n)R^{e}_{\alpha}(x)=\sum^{n}_{i=1}x_{i}\frac {\partial}{\partial x_{i}}R^{e}_{\alpha}(x), \end{gather*} we have that $R^{H}_{\alpha}(x)$ and $R^{e}_{\alpha}(x)$ are homogeneous distributions of order $\alpha-n$. Donoghue \cite[pp. 154-155]{d1} proved that the every homogeneous distribution is a tempered distribution. This completes the proof. \end{proof} \begin{lemma} \label{lem2.4} The convolution $R^{e}_{\alpha}(x)\ast R^{H}_{\alpha}(x)$ exists and is a tempered distribution. \end{lemma} \begin{proof} Choose $\mathop{\rm supp}R^{H}_{\alpha}(x)=K\subset\Gamma_{+}$ where $K$ is a compact set. Then $R^{H}_{\alpha}(x)$ is a tempered distribution with compact support. By Donoghue \cite[pp. 156-159]{d1}, $R^{e}_{\alpha}(x)\ast R^{H}_{\alpha}(x)$ exists and is a tempered distribution. \end{proof} \begin{lemma}[Convolution of $R^{e}_{\alpha}(x)$ and $R^{H}_{\alpha}(x)$] \label{lemma2.5} Let $R^{e}_{\alpha}(x)$ and $R^{H}_{\alpha}(x)$ defined by \eqref{2.5} and \eqref{2.2} respectively, then we obtain the following: \begin{itemize} \item[(1)] $R^{e}_{\alpha}(x)\ast R^{e}_{\beta}(x) =R^{e}_{\alpha+\beta}(x)$ when $\alpha$ and $\beta$ are complex parameters; \item[(2)] $R^{H}_{\alpha}(x)\ast R^{H}_{\beta}(x) =R^{H}_{\alpha+\beta}(x)$ when $\alpha$ and $\beta$ are integers, except when both $\alpha$ and $\beta$ are odd. \end{itemize} \end{lemma} \begin{proof} For the first formula, see \cite[p. 158]{d1}. For the second formula, when $\alpha$ and $\beta$ are both even integers; see \cite{k2}. For the case $\alpha$ is odd and $\beta$ is even or $\alpha$ is even and $\beta$ is odd, by Trione \cite{t1}, we have $$\label{2.8} \square^kR^{H}_{\alpha}(x)=R^{H}_{\alpha-2k}(x)$$ and $$\label{2.9} \square^kR^{H}_{2k}(x)=\delta(x),\quad k=0,1,2,3,\dots$$ where $\square^k$ is the Ultra-hyperbolic operator iterated $k$-times defined by $$\square^k=\Big(\sum^p_{i=1}\frac {\partial^{2}}{\partial x^{2}_{i}}- \sum^{p+q}_{j=p+1}\frac {\partial^{2}}{\partial x^{2}_{j}}\Big)^k.$$ Now let $m$ be an odd integer. We have $\square^kR^{H}_{m}(x)=R^{H}_{m-2k}(x)$ and $$R^{H}_{2k}(x)\ast \square^kR^{H}_{m}(x)=R^{H}_{2k}(x)\ast R^{H}_{m-2k}(x)$$ or \begin{gather*} \left(\square^kR^{H}_{2k}(x)\right)\ast R^{H}_{m}(x)= R^{H}_{2k}(x)\ast R^{H}_{m-2k}(x), \\ \delta \ast R^{H}_{m}(x)=R^{H}_{2k}(x)\ast R^{H}_{m-2k}(x). \end{gather*} Thus $$R^{H}_{m}(x)=R^{H}_{2k}(x)\ast R^{H}_{m-2k}(x).$$ Since $m$ is odd, hence $m-2k$ is odd and $2k$ is a positive even. Put $\alpha=2k,~\beta=m-2k$, we obtain $$R^{H}_{\alpha}(x)\ast R^{H}_{\beta}(x)=R^{H}_{\alpha+\beta}(x)$$ when $\alpha$ is nonnegative even and $\beta$ is odd. For the case when $\alpha$ is negative even and $\beta$ is odd, by \eqref{2.8} we have $$\square^kR^{H}_{0}(x)=R^{H}_{-2k}(x)$$ or $\square^k\delta=R^{H}_{-2k}(x)$, where $R^{H}_{0}(x)=\delta$. Now when $m$ is odd, $$R^{H}_{-2k}(x)\ast \square^kR^{H}_{m}(x)=R^{H}_{-2k}(x)\ast R^{H}_{m-2k}(x)$$ or \begin{gather*} \left(\square^k\delta\right)\ast \square^kR^{H}_{m}(x) = R^{H}_{-2k}(x)\ast R^{H}_{m-2k}(x),\\ \delta \ast\square^{2k}R^{H}_{m}(x)=R^{H}_{-2k}(x)\ast R^{H}_{m-2k}(x). \end{gather*} Thus $$R^{H}_{m-2(2k)}(x)=R^{H}_{-2k}(x)\ast R^{H}_{m-2k}(x).$$ Put $\alpha=-2k$ and $\beta=m-2k$, now $\alpha$ is negative even and $\beta$ is odd. Then we obtain $$R^{H}_{\alpha}(x)\ast R^{H}_{\beta}(x)=R^{H}_{\alpha+\beta}(x).$$ That completes the proof. \end{proof} \section{Main Results} \begin{theorem} \label{thm3.1} Given the equation $$\label{3.1} \oplus^k G(x)=\delta(x),$$ where $\oplus^k$ is the oplus operator iterated $k$ times defined by \eqref{1.8}, $\delta(x)$ is the Dirac-delta distribution, $x\in \mathbb{R}^{n}$, and $k$ is a nonnegative integer. Then $$\label{3.2} G(x)=\left(R^{H}_{6k}(\upsilon)\ast(-1)^{3k}R^{e}_{6k}(x)\right)* \left(C^{*k}(x)\right)^{*-1}$$ is a Green's function or an elementary solution for the operator $\oplus^k$, where $$\label{3.3} C(x)=\frac{1}{2}R^H_{4}(x)+\frac{1}{2}(-1)^2R^e_{4}(x),$$ where $C^{*k}(x)$ denotes the convolution of $C$ with itself $k$ times, $\left(C^{*k}(x)\right)^{*-1}$ denotes the inverse of $C^{*k}(x)$ in the convolution algebra. Moreover $G(x)$ is a tempered distribution. \end{theorem} For a proof of the above theorem, see \cite{s1}. \begin{theorem} \label{thm3.2} For \$0