TY - JOUR

T1 - The neumann problem on lipschitz domains in hardy spaces of order less than one

AU - Brown, Russell M.

PY - 1995/12

Y1 - 1995/12

N2 - Recently, B.E.J. Dahlberg and C.E. Kenig considered the Neumann problem, Δu = 0 in D, ∂u/∂v = f on ∂D, for Laplace’s equation in a Lipschitz domain D. One of their main results considers this problem when the data lies in the atomic Hardy space H1(∂D) and they show that the solution has gradient in L1(∂D). The aim of this paper is to establish an extension of their theorem for data in the Hardy space Hp(∂D), 1-ε < p < 1, where 0 < ε < 1/n is a positive constant which depends only on m, the maximum of the Lipschitz constants of the functions which define the boundary of the domain. We also extend G. Verchota’s and Dahlberg and Kenig’s theorem on the potential representation of solutions of the Neumann problem to the range 1 -ε < p < 1. This has the interesting consequence that the double-layer potential is invertible on Holder spaces Cα (∂D) for a close to zero.

AB - Recently, B.E.J. Dahlberg and C.E. Kenig considered the Neumann problem, Δu = 0 in D, ∂u/∂v = f on ∂D, for Laplace’s equation in a Lipschitz domain D. One of their main results considers this problem when the data lies in the atomic Hardy space H1(∂D) and they show that the solution has gradient in L1(∂D). The aim of this paper is to establish an extension of their theorem for data in the Hardy space Hp(∂D), 1-ε < p < 1, where 0 < ε < 1/n is a positive constant which depends only on m, the maximum of the Lipschitz constants of the functions which define the boundary of the domain. We also extend G. Verchota’s and Dahlberg and Kenig’s theorem on the potential representation of solutions of the Neumann problem to the range 1 -ε < p < 1. This has the interesting consequence that the double-layer potential is invertible on Holder spaces Cα (∂D) for a close to zero.

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U2 - 10.2140/pjm.1995.171.389

DO - 10.2140/pjm.1995.171.389

M3 - Article

AN - SCOPUS:84972578171

SN - 0030-8730

VL - 171

SP - 389

EP - 407

JO - Pacific Journal of Mathematics

JF - Pacific Journal of Mathematics

IS - 2

ER -