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 L2,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 ∂ν = g ∈ H2,λ(∂Ω) on ∂Ω, ‖(∇u)‖H2,λ(∂Ω) < ∞ has a unique solution. Here H2,λ(∂Ω) is an atomic space with the property (H2,λ(∂Ω))∗ = L2,λ(∂Ω). The invertibility of layer potentials on L2,λ(∂Ω) and H2,λ(∂Ω) is also obtained. Finally we study the Dirichlet problem for the biharmonic equation. We establish a similar estimate in L2,λ for the biharmonic equation, in which case the range 0 ≤ λ < 2 + ε is sharp for n = 4 or 5.