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Extrapolation for the Lp Dirichlet Problem in Lipschitz Domains

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Abstract

Let L be a second-order linear elliptic operator with complex coefficients. It is shown that if the Lp Dirichlet problem for the elliptic system L(u) = 0 in a fixed Lipschitz domain Ω in ℝd is solvable for some \(1 < p = {p_0} < \frac{{2\left( {d - 1} \right)}}{{d - 2}},\) then it is solvable for all p satisfying

$${p_0} < p < \frac{{2\left( {d - 1} \right)}}{{d - 2}}+\varepsilon.$$

The proof is based on a real-variable argument. It only requires that local solutions of L(u) = 0 satisfy a boundary Cacciopoli inequality.

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Authors and Affiliations

  1. Department of Mathematics, University of Kentucky, Lexington, Kentucky, 40506, USA

    Zhongwei Shen

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  1. Zhongwei Shen
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Correspondence to Zhongwei Shen.

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Dedicated to Carlos E. Kenig on the Occasion of His 65th Birthday

Supported in part by NSF (Grant No. DMS-1600520)

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Shen, Z. Extrapolation for the Lp Dirichlet Problem in Lipschitz Domains. Acta. Math. Sin.-English Ser. 35, 1074–1084 (2019). https://doi.org/10.1007/s10114-019-8199-6

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