IJPAM: Volume 30, No. 2 (2006)

$L^{2}$ POTENTIAL SPACES IN $\mathbb{R}^{n}$

S.I. Othman
Department of Mathematics
College of Science
King Saud University
P.O. Box 2455, Riyadh, 11451, KINGDOM OF SAUDI ARABIA
e-mail: sadoon@ksu.edu.sa

Abstract.Given an elliptic differential operator $L$ of order 2 with constant coefficients and a domain $\Omega $ in $%
^{n}$. We obtain some sufficient conditions for the existence of solutions for $\left( -L\right) ^{i}u\geq 0$ on $\Omega $, $0\leq i\leq 2$. When such solutions exist on $\Omega $, we prove some interesting results, including the representation of the solutions $h$ of $L\left( Lh\right) =0$ outside a compact set in $\Omega $, by means of certain special functions defined on the whole of $\Omega $.

Received: May 26, 2006

AMS Subject Classification: 35B15, 31D05

Key Words and Phrases: elliptic differential operator $L$ of order 2; $L^{2}-$ potentials

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2006
Volume: 30
Issue: 2