Potential Analysis

, Volume 36, Issue 3, pp 405–428

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

Article

DOI: 10.1007/s11118-011-9233-1

Cite this article as:
Tao, X. Potential Anal (2012) 36: 405. doi:10.1007/s11118-011-9233-1

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 Hp 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.

Keywords

Regularity problem Neumann problem Singular Schrödinger equation Hardy–Sobolev space Lipschitz domain Layer potential 

Mathematics Subject Classifications (2010)

35J10 35J25 42B30 31B25 35B65 

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Department of MathematicsZhejiang University of Science and TechnologyHangzhouChina

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