Layer potentials and boundary value problems for elliptic equations with complex L∞L∞ coefficients satisfying the small Carleson measure norm condition

We consider divergence form elliptic equations Lu:=∇⋅(A∇u)=0Lu:=∇⋅(A∇u)=0 in the half space R+n+1:={(x,t)∈Rn×(0,∞)}, whose coefficient matrix A is complex elliptic, bounded and measurable. In addition, we suppose that A   satisfies some additional regularity in the direction transverse to the boundary, namely that the discrepancy A(x,t)−A(x,0)A(x,t)−A(x,0) satisfies a Carleson measure condition of Fefferman–Kenig–Pipher type, with small Carleson norm. Under these conditions, we establish a full range of boundedness results for double and single layer potentials in LpLp, Hardy, Sobolev, BMO and Hölder spaces. Furthermore, we prove solvability of the Dirichlet problem for L  , with data in Lp(Rn)Lp(Rn), BMO(Rn)BMO(Rn), and Cα(Rn)Cα(Rn), and solvability of the Neumann and Regularity problems, with data in the spaces Lp(Rn)/Hp(Rn)Lp(Rn)/Hp(Rn) and L1p(Rn)/H1,p(Rn) respectively, with the appropriate restrictions on indices, assuming invertibility of layer potentials for the t  -independent operator L0:=−∇⋅(A(⋅,0)∇)L0:=−∇⋅(A(⋅,0)∇).