TY - JOUR

T1 - The mixed problem in L p for some two-dimensional Lipschitz domains

AU - Lanzani, Loredana

AU - Capogna, Luca

AU - Brown, Russell M.

N1 - Funding Information:
L. Lanzani, L. Capogna and R. M. Brown were supported, in part, by the U.S. National Science Foundation.

PY - 2008/9

Y1 - 2008/9

N2 - We consider the mixed problem, {Δ u = 0 in Ω ∂u = f N on N u = fD on D in a class of Lipschitz graph domains in two dimensions with Lipschitz constant at most 1. We suppose the Dirichlet data, f D , has one derivative in L p (D) of the boundary and the Neumann data, f N , is in L p (N). We find a p 0 > 1 so that for p in an interval (1, p 0), we may find a unique solution to the mixed problem and the gradient of the solution lies in L p .

AB - We consider the mixed problem, {Δ u = 0 in Ω ∂u = f N on N u = fD on D in a class of Lipschitz graph domains in two dimensions with Lipschitz constant at most 1. We suppose the Dirichlet data, f D , has one derivative in L p (D) of the boundary and the Neumann data, f N , is in L p (N). We find a p 0 > 1 so that for p in an interval (1, p 0), we may find a unique solution to the mixed problem and the gradient of the solution lies in L p .

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U2 - 10.1007/s00208-008-0223-6

DO - 10.1007/s00208-008-0223-6

M3 - Article

AN - SCOPUS:46649096818

VL - 342

SP - 91

EP - 124

JO - Mathematische Annalen

JF - Mathematische Annalen

SN - 0025-5831

IS - 1

ER -