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Extension Problems Related to the Higher Order Fractional Laplacian

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Abstract

Caffarelli and Silvestre [Comm. Part. Diff. Eqs., 32, 1245–1260 (2007)] characterized the fractional Laplacian (−Δ)s as an operator maps Dirichlet boundary condition to Neumann condition via the harmonic extension problem to the upper half space for 0 < s < 1. In this paper, we extend this result to all s > 0. We also give a new proof to the dissipative a priori estimate of quasi-geostrophic equations in the framework of Lp norm using the Caffarelli–Silvestre’s extension technique.

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Authors and Affiliations

  1. School of Mathematical Sciences, Fudan University, Shanghai, 200433, P. R. China

    Yu Kang Chen

  2. School of Mathematical Sciences and Shanghai Center for Mathematical Sciences, Fudan University, Shanghai, 200433, P. R. China

    Zhen Lei

  3. Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou, 310018, P. R. China

    Chang Hua Wei

Authors
  1. Yu Kang Chen
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  2. Zhen Lei
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  3. Chang Hua Wei
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Correspondence to Zhen Lei.

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Chen, Y.K., Lei, Z. & Wei, C.H. Extension Problems Related to the Higher Order Fractional Laplacian. Acta. Math. Sin.-English Ser. 34, 655–661 (2018). https://doi.org/10.1007/s10114-017-7325-6

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  • DOI: https://doi.org/10.1007/s10114-017-7325-6

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