Monomial ideals of minimal depth

Let S be a polynomial algebra over a field. We study classes of monomial ideals (as for example lexsegment ideals) of S having minimal depth. In particular, Stanley's conjecture holds for these ideals. Also we show that if I is a monomial ideal with Ass(S/I) = {P₁, P₂, ..., P_s} and P_i ⊄ ∑^s_1=j≠i P_j for all i ∊ [s], then Stanley’s conjecture holds for S/I.