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The Regularity Problems with Data in Hardy–Sobolev Spaces for Singular Schrödinger Equation in Lipschitz Domains

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Abstract

Let the nonnegative singular potential V belong to the reverse Hölder class \({\mathcal B}_n\) on \({\mathbb R}^n\), and let (n − 1)/n < p ≤ 2, we establish the solvability and derivative estimates for the solutions to the Neumann problem and the regularity problem of the Schrödinger equation − Δu + Vu = 0 in a connected Lipschitz domain Ω, with boundary data in the Hardy space \(H^p(\partial \Omega)\) and the modified Hardy–Sobolev space \(H_{1, V}^p(\partial \Omega)\) related to the potential V. To deal with the H p regularity problem, we construct a new characterization of the atomic decomposition for \(H_{1, V}^p(\partial \Omega)\) space. The invertibility of the boundary layer potentials on Hardy spaces and Hölder spaces are shown in this paper.

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Authors and Affiliations

  1. Department of Mathematics, Zhejiang University of Science and Technology, Hangzhou, 310023, China

    Xiangxing Tao

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  1. Xiangxing Tao
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Correspondence to Xiangxing Tao.

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This work is funded by the NNSF of China under Grant #10771110 and #10471069.

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Tao, X. The Regularity Problems with Data in Hardy–Sobolev Spaces for Singular Schrödinger Equation in Lipschitz Domains. Potential Anal 36, 405–428 (2012). https://doi.org/10.1007/s11118-011-9233-1

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  • DOI: https://doi.org/10.1007/s11118-011-9233-1

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