The regularity of monomial ideals and their integral closures

Let $I$ be a monomial ideal in a polynomial ring $S=K[x_1,\dots ,x_n]$ over a field $K$ with $n=2$ or $3$, and let $\bar{I}$ be its integral closure. We will show that $\text{reg}(\bar{I})\le \text{reg}(I)$. Furthermore, if $I$ is generated by elements of degree $d$, then $\text{reg}(I)=d$ if and only if $I$ has linear quotients.