Abstract In this paper, we prove an extrapolation result for complex coefficient divergence form operators that satisfy a strong ellipticity condition known as p-ellipticity. Specifically, let Ω be a chord-arc domain in R n and the operator L = ∂ i ( A i j ( x ) ∂ j ) + B i ( x ) ∂ i be elliptic, with | B i ( x ) | ≤ K δ ( x ) − 1 for a small K. Let p 0 = sup ⁡ { p > 1 : A is p -elliptic } . We establish that if the L q Dirichlet problem is solvable for L for some 1 q p 0 ( n − 1 ) ( n − 2 ) , then the L p Dirichlet problem is solvable for all p in the range [ q , p 0 ( n − 1 ) ( n − 2 ) ) . In particular, if the matrix A is real, or n = 2 , the L p Dirichlet problem is solvable for p in the range [ q , ∞ ) .

中文翻译：

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具有复系数的椭圆方程的狄利克雷问题的外推

摘要 在本文中，我们证明了满足称为 p 椭圆度的强椭圆度条件的复系数散度形式算子的外推结果。具体而言，令Ω为R n 中的弦-弧域，且算子L = ∂ i ( A ij ( x ) ∂ j ) + B i ( x ) ∂ i 为椭圆形，其中| B i ( x ) | ≤ K δ ( x ) − 1 对于小 K。令 p 0 = sup ⁡ { p > 1 : A is p -elliptic } 。我们确定，如果 L q Dirichlet 问题对于 L 的某些 1 qp 0 ( n − 1 ) ( n − 2 ) 是可解的，那么 L p Dirichlet 问题对于 [ q , p 0 ( n − 1 ) ( n − 2 ) ) 。特别是，如果矩阵 A 是实数，或 n = 2 ，则 L p Dirichlet 问题对于 [ q , ∞ ) 范围内的 p 是可解的。