Boundary value problems in Morrey spaces for elliptic systems on Lipschitz domains

Let Ω be a bounded Lipschitz domain in [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /], n ≥ 3. Let [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /] be a second order elliptic system with constant coefficients satisfying the Legendre-Hadamard condition. We consider the Dirichlet problem [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /] = 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)*║_{L^2,λ(∂Ω)} ≤ C║f║_{L^2,λ(∂Ω)}. If [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="04i" /] satisfies the strong elliptic condition and 0 ≤ λ < min (n-1, 2+ε), we show that the Neumann type problem [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="05i" /] = 0 in Ω, ∂u/ ∂ν = g ∈ H^2,λ(∂Ω) on ∂Ω, ║(∇u)*║_{H^2,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.