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 ${\mathrm{H}}_I^i(R)\mathrm{\ne }0$, then ${\mathrm{Supp}}_R(D({\mathrm{H}}_I^i(R)))=\mathrm{Spec}(R)$.