TY - JOUR

T1 - The mixed problem in lipschitz domains with general decompositions of the boundary

AU - Taylor, J. L.

AU - Ott, K. A.

AU - Brown, R. M.

PY - 2013

Y1 - 2013

N2 - This paper continues the study of the mixed problem for the Laplacian. We consider a bounded Lipschitz domain Ω ⊂ Rn, n ≥ 2, with boundary that is decomposed as ∂Ω = D ∪ N, with D and N disjoint. We let Λ denote the boundary of D (relative to ∂Ω) and impose conditions on the dimension and shape of Λ and the sets N and D. Under these geometric criteria, we show that there exists p0 > 1 depending on the domain Ω such that for p in the interval (1, p0), the mixed problem with Neumann data in the space Lp(N) and Dirichlet data in the Sobolev space W1,p(D) has a unique solution with the non-tangential maximal function of the gradient of the solution in Lp(∂Ω). We also obtain results for p = 1 when the Dirichlet and Neumann data come from Hardy spaces, and a result when the boundary data comes from weighted Sobolev spaces.

AB - This paper continues the study of the mixed problem for the Laplacian. We consider a bounded Lipschitz domain Ω ⊂ Rn, n ≥ 2, with boundary that is decomposed as ∂Ω = D ∪ N, with D and N disjoint. We let Λ denote the boundary of D (relative to ∂Ω) and impose conditions on the dimension and shape of Λ and the sets N and D. Under these geometric criteria, we show that there exists p0 > 1 depending on the domain Ω such that for p in the interval (1, p0), the mixed problem with Neumann data in the space Lp(N) and Dirichlet data in the Sobolev space W1,p(D) has a unique solution with the non-tangential maximal function of the gradient of the solution in Lp(∂Ω). We also obtain results for p = 1 when the Dirichlet and Neumann data come from Hardy spaces, and a result when the boundary data comes from weighted Sobolev spaces.

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U2 - 10.1090/S0002-9947-2012-05711-4

DO - 10.1090/S0002-9947-2012-05711-4

M3 - Article

AN - SCOPUS:84875524230

SN - 0002-9947

VL - 365

SP - 2895

EP - 2930

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

IS - 6

ER -