TY - JOUR

T1 - Layer potentials and regularity for the Dirichlet problem for Laplace's equation in Lipschitz domains

AU - Verchota, Gregory

PY - 1984/12

Y1 - 1984/12

N2 - For D, a bounded Lipschitz domain in Rn, n ≥ 2, the classical layer potentials for Laplace's equation are shown to be invertible operators on L2(∂D) and various subspaces of L2(∂D). For 1 < p ≤ 2 and data in Lp(∂D) with first derivatives in Lp(∂D) it is shown that there exists a unique harmonic function, u, that solves the Dirichlet problem for the given data and such that the nontangential maximal function of ▽u is in Lp(∂D). When n = 2 the question of the invertibility of the layer potentials on every Lp(∂D), 1 < p < ∞, is answered.

AB - For D, a bounded Lipschitz domain in Rn, n ≥ 2, the classical layer potentials for Laplace's equation are shown to be invertible operators on L2(∂D) and various subspaces of L2(∂D). For 1 < p ≤ 2 and data in Lp(∂D) with first derivatives in Lp(∂D) it is shown that there exists a unique harmonic function, u, that solves the Dirichlet problem for the given data and such that the nontangential maximal function of ▽u is in Lp(∂D). When n = 2 the question of the invertibility of the layer potentials on every Lp(∂D), 1 < p < ∞, is answered.

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U2 - 10.1016/0022-1236(84)90066-1

DO - 10.1016/0022-1236(84)90066-1

M3 - Review article

AN - SCOPUS:48549113823

SN - 0022-1236

VL - 59

SP - 572

EP - 611

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

IS - 3

ER -