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Green's Formulas and Poisson's Equation for Bosonic Laplacians
  • Chao Ding,
  • John Ryan
Chao Ding
Masaryk University
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John Ryan
University of Arkansas Fayetteville
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Peer review status:UNDER REVIEW

29 May 2020Submitted to Mathematical Methods in the Applied Sciences
05 Jun 2020Assigned to Editor
05 Jun 2020Submission Checks Completed
05 Jun 2020Reviewer(s) Assigned

Abstract

A bosonic Laplacian is a conformally invariant second order differential operator acting on smooth functions defined on domains in Euclidean space and taking values in higher order irreducible representations of the special orthogonal group. In this paper, we firstly introduce the motivation for study of the generalized Maxwell operators and bosonic Laplacians (also known as the higher spin Laplace operators). Then, with the help of connections between Rarita-Schwinger type operators and bosonic Laplacians, we solve Poisson’s equation for bosonic Laplacians. A representation formula for bounded solutions to Poisson’s equation in Euclidean space is also provided. In the end, we provide Green’s formulas for bosonic Laplacians in scalar-valued and Clifford-valued cases, respectively. These formulas reveal that bosonic Laplacians are self-adjoint with respect to a given L2 inner product on certain compact supported function spaces.