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/∂v = g ∈ H2,λ (∂Ω) on on, ∂Ω ∥(∇u)* ∂H2,λ(∂Ω) < ∈ has a unique solution. Here H2,λ(∂Ω) is an atomic space with the property (H2,λ(∂Ω))* = L 2,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.
|Number of pages||37|
|Journal||American Journal of Mathematics|
|State||Published - Oct 2003|
ASJC Scopus subject areas
- Mathematics (all)