论文标题
格林的公式和泊松方程
Green's Formulas and Poisson's Equation for Bosonic Laplacians
论文作者
论文摘要
玻色粒拉普拉斯是一个在欧几里得空间中定义的平滑函数上作用于平滑函数的同型不变的二阶差异操作员，并以更高阶段的不可减至的不可约合表示值。在本文中，我们首先介绍了研究通用的麦克斯韦操作员和玻色晶（也称为较高自旋拉普拉斯操作员）的动机。然后，借助Rarita-Schinginger类型操作员与玻色氏拉普拉斯人之间的连接，我们为玻色氏拉普拉斯人解决了Poisson的方程式。还提供了欧几里得空间中泊松方程有限解决方案的表示公式。最后，我们分别在标量价值和克利福德（Clifford）价值案件中为玻色粒拉普拉斯（Bosonic Laplacians）提供了格林的公式。这些公式表明，在某些紧凑的支持功能空间上，玻色粒拉普拉斯与给定的$ l^2 $内部产品是自动的。
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 $L^2$ inner product on certain compact supported function spaces.
