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 of order 2 with constant coefficients and a domain $\Omega$ in $% \mathbb{R} ^{n}$ . 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 of $L\left( Lh\right) =0$ outside a compact set in $\Omega$ , by means of certain special functions defined on the whole of $\Omega$ .