Stanley depth of the integral closure of monomial ideals

Let be a monomial ideal in the polynomial ring S = K[x(1).....x(n)]. We study the Stanley depth of the integral closure (I) over bar of I We prove that for every integer k >= 1 the inequalities (S/(I) over bar (k)) <= sdepth (S/(I) over bar and sdepth ((I) over bar (k)) <= sdepth ((I) over bar) hold. We also prove that for every monomial ideal I subset of S there exist integers k(1), k(2) >= 1 such that for every s >= 1, the inequalities sdepth (S/(I) over bar) <= sdepth (S (I) over bar) and sdepth (I-sk2) <= sdepth ((I) over bar) hol. In particular, and min(k) {sdepth (S/I-k) <= sdepth (S/(I) over bar) and min(k) {sdepth (I) over bark) <= sdepth (I) over bar We conjecture that for every integrally closed monomial ideal I, the inequalities sdepth (S/I) >= n-l (I) and sdepth (I) >= n -l + 1 +hold, where l(1)is the analytic spread of . Assuming the conjecture is true, it follows together with the Burch's inequality that Stanley's conjecture holds for and for I-k and S/I-k for K >> 0 provided that I is a normal ideal.