IJPAM: Volume 72, No. 4 (2011)

ON SOME OPERATOR
RELATED TO TRI-LAPLACE EQUATION

Wanchak Satsanit
Department of Mathematics
Faculty of Science
Maejo University
Chiang Mai, 50290, THAILAND


Abstract. In this paper, we study the operator $\otimes^{k}_{w}$ where $\otimes^{k}_{w}$ is the operator iterated $k$ times and is defined by

\begin{eqnarray*} \otimes^{k}_{w}&=&\left(\left(\sum^{p}_{i=1}\frac{\partial^2... ...+1}\frac{\partial^2}{\partial x^2_j}\right)^{3}\right)^{k},\\ \end{eqnarray*}


where $p+q=n$ is the dimension of the Euclidean space $\mathbb{R}^n.$ At first we study the elementary solution or the Green function of the operator $\otimes^{k}_{w}$ and then such a solution is related to the solution of the Laplace equation. We found that the relationship of such solutions depending on the condition $p, q$ and k. Finally, we applied the elementary solution finding solution equation $\otimes^{k}_{w}u(x)=f(x).$ where $u(x)$ is an unknown function for $x=(x_1,x_2,\ldots,x_n)\in \mathbb{R}^n$, $f(x)$ is the generalized function, $k$ is a positive integer.

Received: June 3, 2011

AMS Subject Classification: generalized function, temper distribution, Laplace equation

Key Words and Phrases: 30Gxx, 35J05

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2011
Volume: 72
Issue: 4