The regularity of monomial ideals and their integral closures

Let I be a monomial ideal in a polynomial ring S=K[x₁,…,xₙ] over a field K with $n=2$ or $3$, and let Ī be its integral closure. We will show that reg (Ī) ≤ reg (I). Furthermore, if I is generated by elements of degree d, then reg (I)=d if and only if I has linear quotients.