On the matlis duals of local cohomology modules

Let $(R,\mathfrak{m})$ be a Noetherian regular local ring containing a field of characteristic $p>0$ and $I$ a nonzero ideal of $R$. In this short note, we prove that if $\operatorname{H}^i_I(R)\neq 0$, then $\operatorname{Supp}_R(D(\operatorname{H}^i_{I}(R)))=\operatorname{Spec}(R)$.