TY - JOUR
T1 - A relationship between the dirichlet and regularity problems for elliptic equations
AU - Shen, Zhongwei
PY - 2007
Y1 - 2007
N2 - Let ℒ = divA∇ be a real, symmetric second order elliptic operator with bounded measurable coefficients. Consider the elliptic equation ℒu = 0 in a bounded Lipschitz domain Ω of ℝn. We study the relationship between the solvability of the Lp Dirichlet problem (D)p with boundary data in Lp(∂Ω) and that of the Lq regularity problem (R)q with boundary data in W1,q(∂Ω), where 1 < p, q < ∞. It is known that the solvability of (R)p implies that of (D)p′. In this note we show that if (D)p′ is solvable, then either (R)p is solvable or (R)q is not solvable for any 1 < q < ∞.
AB - Let ℒ = divA∇ be a real, symmetric second order elliptic operator with bounded measurable coefficients. Consider the elliptic equation ℒu = 0 in a bounded Lipschitz domain Ω of ℝn. We study the relationship between the solvability of the Lp Dirichlet problem (D)p with boundary data in Lp(∂Ω) and that of the Lq regularity problem (R)q with boundary data in W1,q(∂Ω), where 1 < p, q < ∞. It is known that the solvability of (R)p implies that of (D)p′. In this note we show that if (D)p′ is solvable, then either (R)p is solvable or (R)q is not solvable for any 1 < q < ∞.
KW - Dirichlet problem
KW - Elliptic equation
KW - Regularity problem
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U2 - 10.4310/MRL.2007.v14.n2.a4
DO - 10.4310/MRL.2007.v14.n2.a4
M3 - Article
AN - SCOPUS:34547433572
SN - 1073-2780
VL - 14
SP - 205
EP - 213
JO - Mathematical Research Letters
JF - Mathematical Research Letters
IS - 2-3
ER -