## Abstract

Let Ω be a bounded Lipschitz domain in ℝ^{n}, 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.

Original language | English |
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Pages (from-to) | 1079-1115 |

Number of pages | 37 |

Journal | American Journal of Mathematics |

Volume | 125 |

Issue number | 5 |

DOIs | |

State | Published - Oct 2003 |

## ASJC Scopus subject areas

- General Mathematics