Boundary value problems in morrey spaces for elliptic systems on lipschitz domains

Let Ω be a bounded Lipschitz domain in ℝⁿ, n ≥ 3. Let ℒ be a second order elliptic system with constant coefficients satisfying the Legendre-Hadamard condition. We consider the Dirichlet problem ℒu = 0 in Ω, u = f on ∂Ω with boundary data f in the Morrey space L^2,λ(∂Ω). Assume that 0 ≤ λ < 2 + ε for n ≥ 4 where ε > 0 depends on Ω, and 0 ≤ λ ≤ 2 for n = 3. We obtain existence and uniqueness results with nontangential maximal function estimate ∥(u)*∥ _{ℒ2,λ(∂Ω)} ≤ C ∥f∥ _{ℒ2,λ(∂Ω)}. If ℒ satisfies the strong elliptic condition and 0 ≤ λ < min (n-1, 2+ε), we show that the Neumann type problem ℒu = 0 in Ω, ∂u/∂v = g ∈ H^2,λ (∂Ω) on on, ∂Ω ∥(∇u)* ∂_H2,λ(∂Ω) < ∈ has a unique solution. Here H^2,λ(∂Ω) is an atomic space with the property (H^2,λ(∂Ω))* = L ^2,λ(∂Ω). The invertibility of layer potentials on L^2,λ(∂Ω) and H^2,λ(∂Ω) is also obtained. Finally we study the Dirichlet problem for the biharmonic equation. We establish a similar estimate in L^2,λ for the biharmonic equation, in which case the range 0 ≤ λ < 2 + ε is sharp for n = 4 or 5.