On the finiteness properties of local cohomology modules for regular local rings

Let $\frak a$ denote an ideal in a regular local (Noetherian) ring $R$ and let $N$ be a finitely generated $R$-module with support in $V(\frak a)$. The purpose of this paper is to show that all homomorphic images of the $R$-modules $\Ext^j_R(N, H^i_{\frak a}(R))$ have only finitely many associated primes, for all $i, j\geq 0$, whenever $\dim R \leq4$ or $\dim R/ \frak a \leq 3$ and $R$ contains a field. In addition, we show that if $\dim R=5$ and $R$ contains a field, then the $R$-modules $\Ext^j_R(N, H^i_{\frak a}(R))$ have only finitely many associated primes, for all $i, j\geq 0$.